Geometric consequences of intrinsic and extrinsic curvature conditions

Jos´ e Carlos D ´ ıaz Ramos GEOMETRIC CONSEQUENCES OF INTRINSIC AND EXTRINSIC CURVATURE CONDITIONS Jos´ e Carlos D ´ ıaz Ramos GEOMETRIC CONSEQUENCES OF INTRINSIC AND EXTRINSIC CURVATURE CONDITIONS Memoria realizada no Departamento de Xeometr´ıa e Topolox´ıa da Facultade de Matem´aticas, baixo a direcci´on dos profesores J¨urgen Berndt e Eduardo Garc´ıa R´ıo, para obter o Grao de Doutor en Ciencias Matem´aticas pola Universidade de Santiago de Compostela. Leveouse a cabo a s´ua defensa o d´ıa 12 de decembro de 2005 na Facultade de Matem´aticas de dita universidade, obtendo a cualificaci´on de Sobresaliente cum laude. Contents Introduction 1 1 Preliminaries and conventions 3 1.1 Semi–Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Geodesics and the exponential map . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Submanifoldgeometry ............................. 6 1.4 Some special classes of semi–Riemannian manifolds . . . . . . . . . . . . . 8 1.4.1 Two–point homogeneous spaces . . . . . . . . . . . . . . . . . . . . 9 1.4.2 Para–complex space forms . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.3 Einstein manifolds and k–stein manifolds . . . . . . . . . . . . . . . 12 I Geometric consequences of algebraic properties of the curvature tensor 13 2 Algebraic curvature tensors and natural operators 17 2.1 Algebraic curvature tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Decomposition of algebraic curvature tensors . . . . . . . . . . . . . 19 2.2 Natural curvature operators . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 The Jacobi operator . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 The higher order Jacobi operator . . . . . . . . . . . . . . . . . . . 26 2.2.3 The Szab´o operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.4 The skew-symmetric curvature operator . . . . . . . . . . . . . . . 27 3 Four–dimensional Osserman metrics 29 3.1 New examples of Osserman metrics with non–diagonalizable Jacobi operators 31 3.1.1 Some geometrical properties . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Osserman para-Hermitian metrics . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Einstein para-Hermitian structures on Walker manifolds . . . . . . 40 3.2.2 Osserman para–Hermitian structures on Walker manifolds . . . . . 45 3.3 Osserman metrics whose Jacobi operators have two distinct eigenvalues . . 46 3.3.1 Self-duality and anti-self-duality conditions . . . . . . . . . . . . . . 47 3.3.2 Explicit form of self-dual Walker metrics . . . . . . . . . . . . . . . 49 v 3.3.3 Proof of Theorem 3.22 . . . . . . . . . . . . . . . . . . . . . . . . . 51 Open problems 55 II Curvature invariants of geodesic spheres and geodesic celestial spheres 57 4 The Riemannian setting 63 4.1 Tubes and Jacobi vector field theory . . . . . . . . . . . . . . . . . . . . . 64 4.2 Geodesic spheres in Riemannian manifolds . . . . . . . . . . . . . . . . . . 66 4.2.1 Curvature and Weyl invariants . . . . . . . . . . . . . . . . . . . . . 67 4.2.2 Total scalar curvatures of geodesic spheres . . . . . . . . . . . . . . 77 4.2.3 Homogeneity and two–point homogeneous spaces . . . . . . . . . . 82 4.3 Applications................................... 85 4.3.1 Total scalar curvatures of geodesic spheres . . . . . . . . . . . . . . 86 4.3.2 Total scalar curvatures of boundaries of geodesic disks . . . . . . . 89 5 Geodesic celestial spheres in Lorentzian manifolds 95 5.1 Volume comparison results . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.1.1 Truncated light cones . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.1.2 Compact distance wedges . . . . . . . . . . . . . . . . . . . . . . . 98 5.1.3 SCLVsets................................ 100 5.1.4 Geodesic celestial spheres . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Volume of geodesic celestial spheres . . . . . . . . . . . . . . . . . . . . . . 103 5.2.1 Power series expansions . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2.2 Volume comparison theorems . . . . . . . . . . . . . . . . . . . . . 109 5.2.3 Characterization of locally isotropic Lorentzian manifolds . . . . . . 111 5.3 Total curvatures of geodesic celestial spheres . . . . . . . . . . . . . . . . . 112 Open problems 119 III Real hypersurfaces in the complex hyperbolic space 121 6 Cohomogeneity one actions on the complex hyperbolic space 127 6.1 Preliminaries .................................. 127 6.2 The complex hyperbolic space as a solvable Lie group . . . . . . . . . . . . 130 6.3 Cohomogeneity one actions on CHn...................... 133 6.3.1 Cohomogeneity one actions with one totally geodesic singular orbit 135 6.3.2 Cohomogeneity one actions with no singular orbits . . . . . . . . . 137 6.3.3 Cohomogeneity one actions with one non–totally geodesic singular orbit................................... 140 7 Real hypersurfaces with constant principal curvatures 163 7.1 Formulas for constant principal curvatures . . . . . . . . . . . . . . . . . . 164 7.2 ProofofTheorem7.1.............................. 167 7.2.1 Principal curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.2.2 Equidistant hypersurfaces and rigidity . . . . . . . . . . . . . . . . 182 Open problems 189 Resumo en galego 193 Bibliography 201 Introduction When studying the geometric properties of a semi–Riemannian manifold, the starting point usually comes from some invariants of the metric structure. Among those invariants, the curvature tensor is perhaps the simplest and most natural one. In the words of R. Osserman [109] The notion of curvature is one of the central concepts of differential geometry; one could argue that it is the central one, distinguishing the geometrical core of the subject from those aspects that are analytical, algebraic or topological. In the words of M. Berger, curvature is the “Number 1 Riemannian invariant and the most natural. Gauss and then Riemann saw it instantly”. Curvature, however, can be studied from several points of view. On the one hand, an essential problem in differential geometry is to relate properties of the curvature tensor to the underlying geometry of the manifold. Another point of view is to consider different kinds of objects naturally associated with the metric structure of the manifold and relate the curvature of the manifold to the properties of these natural constructions. When dealing with a complicated object such as the curvature tensor, it is interesting to decompose it in more elementary constituents. Usually, these smaller parts give a simplified picture and a deeper insight into the whole problem. In Chapter 2 we show that the curvature tensor may be decomposed in terms of some simple algebraic curvature tensors. This is of special importance when considering Osserman–like problems. Furthermore, the fact that the whole curvature tensor is very difficult to handle derived the investigation to the consideration of geometric objects naturally associated with the curvature. Typical examples are the sectional curvature, the Ricci tensor or the scalar curvature. Part I of this thesis fits into this philosophy. Among the different operators that can be defined from the curvature tensor, we are specially interested in the Jacobi operator, which encodes important geometric information and whose properties strongly influence the underlying geometry of the manifold. The Jacobi operator and Jacobi vector field theory are important tools in semi–Riemannian geometry. They provide a good description of curvature, behavior of geodesics and geometry of certain kinds of submanifolds. Thus, understanding the Jacobi operator of a semi–Riemannian manifold allows us to characterize the geometry of the manifold in several cases. Chapter 3 of this thesis is devoted to the investigation of the Jacobi operator in relation to the so–called Osserman problem. In Chapter 3 we focus on the Osserman problem in dimension four. Our main goal is to 1 8 1 Preliminaries and conventions We write ²=g(ξ, ξ)∈ {−1,1}. Hence the second fundamental form II is a multiple of ξ. We define the scalar second fundamental form σof Mby the equality II(X, Y ) = ² σ(X, Y )ξ for X, Y ∈Γ(TM), that is, σ(X, Y ) = g(II(X, Y ), ξ). We denote by S=Sξthe shape operator with respect to ξ. With respect to the scalar second fundamental form we have g(SX, Y ) = σ(X, Y ). The Gauss formula and the Weingarten equation can be written as ∇XY=¯ ∇XY+² g(SX, Y )ξand ¯ ∇Xξ=SX. Then, the Gauss and Codazzi equation reduce to ¯ RXY V W =RXY V W −² g(SX, V )g(SY, W) + ² g(SX, W)g(SY, V ), (∇XS)Y−(∇YS)X=−¯ RXY ξ, whereas the Ricci equation does not give further information for hypersurfaces. The mean curvature vector His proportional to the vector ξ. We define the scalar mean curvature hby the equation H=h ξ. We say that λ:M→Ris a principal curvature of M(associated with ξ) if there exists a vector field X∈Γ(TM) such that SX =λX. If ¯ Mis a Riemannian manifold, the shape operator Sis diagonalizable at every point because it is a self–adjoint map and the metric is positive definite. If λis a principal curvature we denote by Tλ(p) the eigenvector space of λ(p) and call it the principal curvature space associated with λ(p). If X∈Tλ(p), X6= 0 we say that X is a principal curvature vector of λat p. We emphasize here that, in general, the principal curvature spaces associated with a principal curvature λdo not always have the same dimension. A connected hypersurface is said to have constant principal curvatures if the shape operator is diagonalizable and its eigenvalues are the same at every point. In this case the principal curvature spaces associated with an eigenvalue λhave the same dimension at any point. We denote by mλthe dimension of any of the vector spaces Tλ(p) and call this number the multiplicity of λ. By Tλwe denote the distribution on Mformed by the principal curvature spaces of λand by Γ(Tλ) we denote the set of all sections of Tλ, that is, the vector fields X∈Γ(TM) such that SX =λX. 1.4 Some special classes of semi–Riemannian manifolds We introduce a few kinds of manifolds which will be of special relevance in this thesis. The description is not intended to be thorough and we restrict ourselves to those types which are going to be used later. A wider study of structures on manifolds can be found for example in [134]. 1.4.1 Two–point homogeneous spaces 9 1.4.1 Two–point homogeneous spaces A connected Riemannian manifold Mis called two–point homogeneous if the isometry group of Macts transitively on equidistant pairs of points. This means that for any p1, p2, q1, q2∈Mwith d(p1, q1) = d(p2, q2), where dis the Riemannian distance function of M, there is an isometry Φ of Msuch that Φ(p1) = p2and Φ(q1) = q2. This definition clearly implies that a two–point homogeneous space is homogeneous and complete. Let Mbe a semi–Riemannian manifold and p∈M. The manifold Mis said to be isotropic at pif the isotropy group of the isometry group of Mat pacts transitively on the unit pseudo–sphere bundle. The manifold is called isotropic if it is isotropic at every point, or equivalently, if for each point p∈Mand any non–null vectors x, y ∈TmMwith g(x, x) = g(y, y) there exists an isometry Φ of Msuch that Φ(p) = pand Φ∗p(x) = Φ∗p(y). The notion of locally isotropic manifold can be defined in an analogous way. If Mis a Riemannian manifold, then Mis two–point homogeneous if and only if it is isotropic. Any two–point homogeneous space is symmetric [122]. Indeed, a simply connected two–point homogeneous space is a flat space, an irreducible symmetric space of rank one or one of its non–compact duals. Hence, a simply connected two–point homogenous space is isometric to one of the following manifolds: (i) The Euclidean space Rn. (ii) The sphere Sn=SO(n+1)/SO(n), the real projective space RPn=SO(n+1)/O(n) or the real hyperbolic space RHn=SO0(1, n)/SO(n). (iii) The complex projective space CPn=SU(n+ 1)/U(n) or the complex hyperbolic space CHn=SU(1, n)/S(U(1)U(n)). (iv) The quaternionic projective space HPn=Sp(n+ 1)/Sp(1)Sp(n) or the quaternionic hyperbolic space HHn=Sp(1, n)/Sp(1)Sp(n). (v) The Cayley projective plane OP2=F4/Spin(9) or the Cayley hyperbolic plane OH2=F−20 4/Spin(9). The examples in (ii) are called real space forms, the examples in (iii) are called complex space forms and the examples in (iv) are called quaternionic space forms. These three constructions can be generalized to the general semi–Riemannian setting. We briefly describe them in what follows. Indefinite real space forms A semi–Riemannian manifold (Mn, g) of signature (r, s) is called a real space form if (M, g) has constant sectional curvature. If (M, g) is a real space form of constant curvature λ∈R, the curvature tensor of (M, g) is given by Rxyz=λ¡g(x, z)y−g(y, z)x¢, 10 1 Preliminaries and conventions for all x, y, z ∈TM. A complete and simply connected real space form is isometric to RPn s=SO(s, r + 1)/O(s, r),RHn s=SO0(s+ 1, r)/SO(s, r) or Rn s according to whether the sectional curvature is positive, negative or zero [133]. Indefinite complex space forms Let (M, J) be an almost complex manifold with almost complex structure J, that is, J is a (1,1)–tensor field on Msatisfying J2=−Id. A semi–Riemannian metric tensor gof signature (2r, 2s) is said to be Hermitian if g(JX, Y )+g(X, JY ) = 0 for all X, Y ∈Γ(TM). If the metric tensor is integrable, that is, if [J, J] = 0 where [J, J](X, Y )=[JX, JY ]− J[JX, Y ]−J[X, JY ]−[X, Y ], then Jis said to be a complex structure. The triple (M2n, g, J) is said to be a K¨ahler manifold if Jis a complex structure and the 2–form Ω(X, Y ) = g(X, JY ) is closed. This couple of conditions can be equivalently described by ∇J= 0, where ∇is the Levi–Civita connection of g. A plane πis called holomorphic if it remains invariant under the complex structure (Jπ ⊂π), and the holomorphic sectional curvature is defined as the restriction of the sectional curvature to non–degenerate holomorphic planes. A K¨ahler manifold (M, g, J) is called a complex space form if (M, g, J) is of constant holomorphic sectional curvature. If this constant is µ, then the curvature tensor of (M, g, J) is given by, Rxyz=µ 4³g(x, z)y−g(y, z)x+g(Jx, z)Jy −g(Jy, z)Jx + 2g(Jx, y)Jz´ for all x, y, z ∈TM. Let z∈TM be a unit vector. The Jacobi operator of zis given by Rz=(µ g(z, z) Id,if z∈RJz, µ 4g(z, z) Id,if z∈Cz⊥. The model spaces of non–zero constant holomorphic sectional curvature are given by the symmetric spaces CPn s=SU(s, r + 1)/U(s, r) and CHn s=SU(s+ 1, r)/S(U(s+ 1)U(r)). Indefinite quaternionic space forms An almost quaternionic manifold is a manifold Mequipped with a 3–dimensional vector bundle Qof (1,1)–tensor fields on Msuch that there exists a local basis {J1, J2, J3}of Qsatisfying J2 i=−Id, i= 1,2,3, and JiJj=Jk, where (i, j, k) is a cyclic permutation of (1,2,3). Such a local basis {J1, J2, J3}is called a canonical local basis of Qand Qis referred to as an almost quaternionic structure on M. A semi–Riemannian metric tensor gof signature (4r, 4s) is said to be adapted to the almost quaternionic structure Qif g(φX, Y ) + g(X, φY ) = 0 for all φ∈Qand X, Y ∈Γ(TM). 1.4.2 Para–complex space forms 11 Let (M, g, Q) be an almost quaternionic manifold and {J1, J2, J3}be a canonical local basis of Q. For each i∈ {1,2,3}, we put Φi(X, Y ) = g(X, JiY), where X, Y ∈Γ(TM). Then, Φiis a locally defined 2–form such that Ω = Φ1∧Φ1+ Φ2∧Φ2+ Φ3∧Φ3gives rise to a globally defined 4–form on M. A quaternionic metric structure (g, Q) is said to be K¨ahler if Ω is parallel (or equivalently, if Qis parallel) with respect to the Levi–Civita connection ∇of g. Let (M, g, Q) be a quaternionic K¨ahler manifold. Then Mhas signature (4r, 4s). Any vector x∈TpMdetermines a 4–dimensional subspace Q(x) = Rx⊕RJ1x⊕RJ2x⊕RJ3x which remains invariant under the action of the quaternionic structure. We call it the Q–section determined by x. If the sectional curvature of planes in Q(x) is a constant ν(x), where x∈TM is non–null, we call this constant ν(x) the quaternionic sectional curvature of (M, g) with respect to x. A quaternionic K¨ahler manifold (M, g, Q) is called a quaternionic space form if (M, g, Q) is of constant quaternionic sectional curvature. Then its curvature tensor is given by Rxyz=ν 4ng(x, z)y−g(y, z)x+ 3 X i=1 ³g(Jix, z)Jiy−g(Jiy, z)Jix+ 2g(Jix, y)Jiz´o, for all x, y, z ∈TM, and where {J1, J2, J3}is a canonical local basis of Q. A non–flat quaternionic space form is isometric to one of the following symmetric spaces HPn s=Sp(s, r + 1)/Sp(1)Sp(s, r) or HHn s=Sp(s+ 1, r)/Sp(1)Sp(s, r). 1.4.2 Para–complex space forms In addition to the well–known examples of semi–Riemannian manifolds described above, there are some other examples which have no Riemannian analog. However, they may be considered as a kind of real version of complex manifolds. Apara–K¨ahler manifold is a symplectic manifold locally diffeomorphic to a product of Lagrangian submanifolds. Such a product induces a decomposition of the tangent bundle TM into a Whitney sum of Lagrangian subbundles Land L0, that is, TM =L⊕L0. By generalizing this definition, an almost para–Hermitian manifold is defined to be an almost symplectic manifold (M, Ω) whose tangent bundle splits into a Whitney sum of Lagrangian subbundles. This implies that the (1,1)–tensor field Jdefined by J=σL−σL0 is an almost para–complex structure (J2= Id) on Msuch that Ω(JX, JY ) = −Ω(X, Y ) for all X, Y ∈Γ(TM), where σLand σ0 Lare the projections of TM onto Land L0, respectively. The 2–form Ω induces a non–degenerate (0,2)–tensor field gon Mdefined by g(X, Y ) = Ω(X, JY ), where X, Y ∈Γ(TM). Now, by using the relation between the almost para– complex and the almost symplectic structures on M, it follows that gdefines a semi– Riemannian metric tensor of signature (n, n) on Mand g(JX, Y ) + g(X, JY ) = 0, where X, Y ∈Γ(TM). The special significance of the para–K¨ahler condition is equivalently stated in terms of the parallelizability of the para–complex structure with respect to the Levi–Civita connection of g, that is, ∇J= 0 [35]. 12 1 Preliminaries and conventions A plane πis called para–holomorphic if it is left–invariant by the action of the para– complex structure J, that is, Jπ ⊂π. The para–holomorphic sectional curvature is defined by the restriction of the sectional curvature to para–holomorphic non–degenerate planes. A para–K¨ahler manifold (M, g, J) is called a para–complex space form if (M, g, J) is of constant para–holomorphic sectional curvature. Hence, the curvature tensor of (M, g, J) is determined by Rxyz=µ 4³g(x, z)y−g(y, z)x−g(Jx, z)Jy +g(Jy, z)Jx −2g(Jx, y)Jz´, for all x, y, z ∈TpMand some constant µ∈R. Then, the Jacobi operator with respect to a unit vector z∈TM is given by Rz=(µ g(z, z) Id,if z∈RJz, µ 4g(z, z) Id,if z∈(Rz⊕RJz)⊥. Non–flat complete and simply connected para–complex space forms are isometric to the symmetric spaces SL(n, R)/SL(n−1,R)×R. 1.4.3 Einstein manifolds and k–stein manifolds A semi–Riemannian manifold (Mn, g) is called an Einstein manifold if the Ricci tensor is proportional to the metric, that is, if there exists a constant λ∈Rsuch that ρ=λ g. Taking traces we easily see that λ=τ/n and hence the scalar curvature is constant. If n > 2 and there exists a function f:M→Rsuch that ρ=f g then, the Schur lemma implies that fis constant and thus the manifold is Einstein. If a semi–Riemannian manifold M has dimension 2 or 3 then, Mis Einstein if and only if Mhas constant sectional curvature. A semi–Riemannian manifold is said to be k–stein, for k≥1, if there exists a constant λsuch that tr Rk x=λg(x, x)kfor all x∈TM, where Rk xis the k–power of the Jacobi operator. Note that a manifold is 1–stein if and only if it is Einstein. We are specially interested in the 2–stein condition, which plays an important role in Part II. With respect to an orthonormal basis {ei}the 2–stein condition may be written as n X i,j=1 g(ei, ei)g(ej, ej)R2 xeixej=λ g(x, x)2. A manifold Mis said to be super–Einstein if n X i,j,k=1 g(ei, ei)g(ej, ej)g(ek, ek)R2 xeiejek=µ g(x, x), for some constant µ. It was shown in [33] that 2–stein manifolds are super–Einstein although the converse is not true. For instance, irreducible symmetric spaces are super– Einstein but not necessarily 2–stein. Part I Geometric consequences of algebraic properties of the curvature tensor 13 A central problem in differential geometry is to relate algebraic properties of the curvature tensor to the underlying geometry of the manifold. From an algebraic point of view the space of algebraic curvature tensors on an n–dimensional vector space Vis a vector space R(V) of dimension n2(n2−1)/12, which makes it very difficult to manipulate. Hence, the investigation focused many times on trying to find suitable bases or sets of generators allowing some simplifications. A typical example is the Singer–Thorpe basis in dimension four (see also [92] for higher dimensions). Recently, the work of B. Fiedler [59] and P. Gilkey [68] showed the existence of nice sets of generators of R(V) constructed from symmetric and skew–symmetric bilinear forms, which seems to be useful in understanding some curvature conditions. Our approach to this problem, based on the use of the Nash embedding theorem and the possibility of realizing geometrically any algebraic curvature tensor, has two main advantages. The first one is that it allows us to obtain some sharper (although not optimal) estimates for the number of generators of R(V). Secondly, it shows that each algebraic curvature tensor can also be seen from an extrinsic point of view as the second fundamental form of a suitable embedding. All these discussions are carried out in Chapter 2. Another purpose of this part is to study the influence of algebraic properties of natural operators associated with the curvature tensor on the manifold geometry. More precisely, our attention is mainly devoted to the investigation of the Jacobi operator by focusing on the structure of four–dimensional Osserman metrics. A semi–Riemannian manifold is said to be Osserman if the eigenvalues of the Jacobi operators are independent of the direction and the base point. Since the group of local isometries of an isotropic space acts transitively on the unit pseudo–sphere bundles, it is clear that any isotropic space is Osserman. No other examples may exist in the Riemannian (dim 6= 16) and Lorentzian settings but there exist non–symmetric and even non–locally homogeneous Osserman metrics in any signature (p, q) with p, q ≥2. Four–dimensional Osserman metrics are of particular interest. First of all, four is the first non–trivial dimension to be considered in the investigation of the Osserman problem (note that any Osserman metric is Einstein, and thus of constant sectional curvature in 15 dimensions 2 and 3), and moreover, four is the lowest possible dimension which supports metrics of neutral non–Lorentzian signature, where the first non–symmetric Osserman metrics were discovered. Due to curvature identities, for any non–null vector x∈TM, the Jacobi operator acts as a self–adjoint operator in x⊥, which has induced metric of Lorentzian signature in the (2,2) setting. Osserman metrics with diagonalizable Jacobi operators have been characterized by N. Blaˇzi´c, N. Bokan and Z. Raki´c [21], who also showed the non–existence of Osserman metrics in dimension four whose Jacobi operators have complex eigenvalues. However, the Lorentzian signature of x⊥supports two other possibilities corresponding to a double or triple root of the minimal polynomial of the Jacobi operators. The fact that all known examples in those situations have nilpotent Jacobi operators and that four– dimensional symmetric Osserman spaces have diagonalizable or two–step nilpotent Jacobi operators motivated a conjecture that Osserman metrics whose Jacobi operators are not diagonalizable must have nilpotent Jacobi operators. Our purpose in Chapter 3 is to answer the above conjecture in the negative by showing explicit examples of Osserman metrics whose Jacobi operators are neither diagonalizable nor nilpotent. Finally, a complete description of such metrics is given in Section 3.3. 16 Chapter 2 Algebraic curvature tensors and natural operators In this chapter we discuss some algebraic properties of the curvature tensor and its covariant derivatives. When studying curvature it is sometimes convenient to work in the algebraic setting. This often simplifies calculations and allows one to distinguish between purely geometric or topological properties and those properties which are imposed by the linear nature of most of the objects that can be defined in a manifold. Section 2.1 is devoted to the study of algebraic curvature tensors. Geometric realizability turns this concept into a very powerful notion when studying manifolds where the curvature tensor verifies some algebraic property. Hence, it is interesting to be capable of decomposing the curvature tensor into more elementary parts which can be studied in an easier way. Theorems 2.3 and 2.4 contribute to this philosophy giving somehow an upper bound of the complexity of the curvature tensor. Some other results are given in relation to this decomposition of the curvature tensor. Section 2.2 deals with certain natural operators that can be defined from the curvature tensor of a semi–Riemannian manifold. We give the basic definitions and results that will be used in the following chapter. 2.1 Algebraic curvature tensors Let Vbe an n-dimensional vector space with an inner product g. An algebraic curvature tensor is a tensor F∈ ⊗4(V∗) satisfying the algebraic identities of the Riemannian curvature tensor, that is, F(x, y, v, w) = −F(y, x, v, w) = −F(x, y, w, v) = F(v, w, x, y), F(x, y, v, w) + F(y, v, x, w) + F(v, x, y, w) = 0. Let us denote by R(V) the vector space of algebraic curvature tensors of V. This vector space has dimension n2(n2−1)/12. 17 24 2 Algebraic curvature tensors and natural operators Finally assume α1, α26= 0 and α3= 0. We show that it is not possible to express the given algebraic curvature tensor as F=γFφ. On the contrary, assume this can be achieved for certain γand φ. Since α1, α26= 0 we have F6= 0 and hence γ6= 0. Then F=γFφ implies φ11φ22 −φ2 12 = 0, φ11φ33 −φ2 13 =α2 2γ, φ22φ33 =α1 2γ, φ11φ23 =φ13φ12, φ12φ33 =φ13φ23, φ12φ23 =φ13φ22. Straightforward calculations show that the above system of equations has no solution. Nevertheless, it is possible to write F=Fφ1+Fφ2. For example take φ1=  0 0 0 0 1 0 0 0 α1 2  , φ2=  1 0 0 0 0 0 0 0 α2 2   and the equality follows after a simple calculation. Corollary 2.10. We have µ(2) = 1 and µ(3) = 2. Proof. First, we observe that for any two–dimensional manifold, the curvature tensor is expressed in terms of the Ricci tensor and thus, any algebraic curvature tensor on a two– dimensional vector space is completely determined by exactly one Fφ. The second assertion is an immediate consequence of Proposition 2.9. Remark 2.11.Theorem 2.3 provides a criteria for the non–existence of embeddings of a given manifold into a Euclidean space. For instance, no Riemannian 3–dimensional manifold whose curvature tensor is as in Proposition 2.9 (b) at some point can be isometrically embedded as a hypersurface in a flat space. 2.2 Natural curvature operators When investigating algebraic properties of the curvature tensor, one usually focus on different kinds of natural operators defined from the curvature, with special attention to their spectrum. Among those operators, the Jacobi operator is probably the most natural and widely investigated. Nevertheless, many interesting information is encoded by other operators such as the Szab´o operator or the skew–symmetric curvature operator. We recall the definitions and some relevant results related to the associated Osserman–like problems. 2.2.1 The Jacobi operator Let Mbe a semi–Riemannian manifold of signature (p, q) and dimension n=p+q. Let S+(M) be the bundle of unit spacelike tangent vectors and S−(M) the bundle of unit timelike tangent vectors. Also, S(M) is defined by Sp(M) = S+ p(M)∪S− p(M) for all p∈M. 2.2.1 The Jacobi operator 25 We recall that the Jacobi operator Rxfor x∈TM is the self-adjoint endomorphism of x⊥ characterized by the identity g(Rx(y), z) = R(x, y, x, z). One says that Mis spacelike Osserman (resp. timelike Osserman) if the eigenvalues of the Jacobi operator are constant on S+(M) (resp. S−(M)). It turns out that these two notions are equivalent and such a manifold is simply said to be Osserman. A manifold M is said to be pointwise Osserman if the eigenvalues of the Jacobi operator are independent of the direction, although they may change from point to point. A manifold is pointwise Osserman if and only if it is k–stein for all k≥1. In particular, every Osserman manifold is Einstein. The local isometries of any isotropic space act transitively on the unit pseudo–sphere bundles and thus the eigenvalues of the Jacobi operator are constant on S(M), which shows that Mis Osserman. R. Osserman [109] wondered whether the converse holds. This question has been called the Osserman conjecture by subsequent authors. This conjecture has been answered in the affirmative in the Riemannian setting if n6= 16 by the work of Q. S. Chi [34] and Y. Nikolayevsky [104], [105], [106]. In the Lorentzian setting (p= 1), an Osserman manifold has constant sectional curvature [19], [61]. In the higher signature setting (p > 1, q > 1) the situation is much more complicated since many non–symmetric examples exist [64]. See for example [62], [68] and the references therein for more information. Moreover, the fact that the spectrum does not completely determine a self–adjoint operator in the indefinite setting suggested the consideration of the Jordan normal form rather than just the eigenvalue structure. Then, one says that (M, g) is spacelike Jordan–Osserman (resp. timelike Jordan–Osserman) if the Jordan normal form of the Jacobi operator is constant on S+(M) (resp. S−(M)). These two notions are not equivalent if n≥5. The structure of a Jordan–Osserman algebraic curvature tensor strongly depends on the signature (p, q) of the metric tensor. Indeed, it has been shown in [71] that the spacelike Jacobi operators of a spacelike Jordan–Osserman algebraic curvature tensor are necessarily diagonalizable whenever p<q, but they can be arbitrarily complicated in the neutral case (p=q) [70]. Example 2.12.[41] Let (~x, ~y) for ~x = (x1, ..., xp) and ~y = (y1, ..., yp) be coordinates on R2p where p≥3. Let f:Rp→Rbe a differentiable function. We define a semi–Riemannian metric gfof signature (p, p) on R2pby gfµ∂ ∂xi,∂ ∂xj¶=∂f ∂xi·∂f ∂xi, gfµ∂ ∂yi,∂ ∂yj¶= 0 and gfµ∂ ∂xi,∂ ∂yj¶=δij . Let φbe the Euclidean Hessian φµ∂ ∂xi,∂ ∂xj¶=∂2f ∂xi∂xj, φµ∂ ∂yi,∂ ∂yj¶= 0 and φµ∂ ∂xi,∂ ∂yj¶=φµ∂ ∂yj,∂ ∂xi¶= 0. Then, the curvature tensor of gfis R=Fφ. We assume that the restriction of φto span{∂/∂xi}is positive definite henceforth. Then Mis a complete semi–Riemannian 26 2 Algebraic curvature tensors and natural operators manifold that is spacelike and timelike Jordan–Osserman. Similarly define φ1by the trivial bilinear extension of φ1µ∂ ∂xi,∂ ∂xj,∂ ∂xk¶=∂3f ∂xi∂xj∂xk. One has ∇R=Fφ,φ1. Thus if fis not quadratic, Mis not a locally symmetric space. With a bit more work one can show that for such a generic f,Mis curvature homogeneous but not locally affine homogeneous. We refer to [52], [75] for further details. 2.2.2 The higher order Jacobi operator Let (M, g) be a semi–Riemannian manifold and let Grr,s(TpM) be the Grassmannian of all subspaces E⊂TpMsuch that the restriction of gto Eis a non–degenerate inner product of signature (r, s). Let {ei}be an orthonormal basis of E∈Grr,s(TpM). We define the higher order Jacobi operator by J(E) = r+s X i,j=1 ²eiRei. where ²i=g(ei, ei). A semi–Riemannian manifold (M, g) is said to be (r, s)–Osserman at p∈Mif the coefficients of the characteristic polynomial of J(E) are independent of E∈Grr,s(TpM) (see [68], [78], [120] and the references therein). An interesting observation is that only the value r+sis important in the previous definition. Moreover, any k– Osserman manifold is of constant sectional curvature in the Riemannian (for k > 1) and Lorentzian settings (see [69], [79]). Again, the situation is more complex in the higher signature case, where many non–symmetric k–Osserman metrics exist (see for example [22], [68]). 2.2.3 The Szab´o operator There is an analogous operator to the Jacobi operator which is defined for ∇R. The Szab´o operator J1(x) is the self–adjoint endomorphism of TM characterized by g(J1(x)y, z) = (∇R)(x, x, y, x, z) = (∇xR)(x, y, x, z). One says that Mis spacelike Szab´o (resp. timelike Szab´o) if the eigenvalues of J1(·) are constant on S+(M) (resp. S−(M)). These notions are equivalent and such a manifold is simply said to be Szab´o. The notion spacelike Jordan–Szab´o (resp. timelike Jordan–Szab´o) is defined similarly. In his study of 2–point homogeneous spaces, Z. I. Szab´o [122] gave a topological argument showing that any Riemannian Szab´o manifold is necessarily a locally symmetric space, that is, ∇R= 0. This result was subsequently extended to the Lorentzian case [79]. In the higher signature setting, the situation is unclear again. The metric gfdescribed in Example 2.12 defines a Szab´o semi–Riemannian manifold of signature (p, p). 2.2.4 The skew-symmetric curvature operator 27 Even in the algebraic setting, there are no known non–zero elements F1∈ R1(V) which are spacelike Jordan–Szab´o. It has been shown in [73] that if F1is a spacelike Jordan– Szab´o algebraic covariant derivative curvature tensor on a vector space of signature (p, q), where q≡1 (mod 2) and p < q or where q≡2 (mod 4) and p < q −1, then F1= 0. This algebraic result yields an elementary proof of the geometrical fact that any pointwise totally isotropic semi–Riemannian manifold with such a signature is locally symmetric. The general question of finding non–trivial spacelike Jordan Szab´o covariant algebraic curvature tensors or showing non–existence remains open. 2.2.4 The skew-symmetric curvature operator Let {e1, e2}be an orthonormal basis for an oriented spacelike (resp. timelike) 2–plane π. The skew–symmetric curvature operator ˜ R(π) is characterized by the identity g(˜ R(π)y, z) = R(e1, e2, y, z). This definition is independent of the particular choice of orthonormal basis. One says that Mis spacelike Ivanov–Petrova (resp. timelike Ivanov–Petrova) if the eigenvalues of ˜ R(·) are constant on the Grassmannian of oriented spacelike (resp. timelike) 2–planes. These two notions are equivalent and such a manifold is simply said to be Ivanov–Petrova. The notions spacelike Jordan–Ivanov–Petrova and timelike Jordan Ivanov–Petrova are defined similarly and are not equivalent. The Riemannian Ivanov–Petrova manifolds have been classified in [74], [107]. They have also been classified in the Lorentzian setting [135] if n≥10. For all these manifolds, the curvature tensors have the form R=Fφwhere φis an idempotent isometry and ˜ R(π) has rank 2. Conversely, in the algebraic setting, if Ris a spacelike Jordan–Ivanov– Petrova algebraic curvature tensor on a vector space of signature (p, q) where q≥5 and where rank ˜ R(·) = 2, then there exist λand φsuch that R=λFφ. The situation in the indefinite setting is again quite different. There exist spacelike Ivanov–Petrova manifolds of signature (p, 2p) where ˜ R(π) has rank 4 and where the curvature tensor does not have the form R=Fφ. We refer to [76] for further details. 28 Chapter 3 Four–dimensional Osserman metrics Any Osserman metric is Einstein and thus of constant sectional curvature in dimension two and three. Therefore, dimension four is the lowest dimensional non–trivial case to study Osserman metrics. Moreover, it supports metrics of neutral signature (2,2) where the Jacobi operator exhibits a completely different behavior with respect to both the Riemannian and Lorentzian settings and enjoys some special features of four–dimensional geometry. Considering the curvature tensor Ras an endomorphism of Λ2(M), we have the following O(2,2)–decomposition for (2,2)–metrics R=τ 12 IdΛ2+ρ0+W: Λ2→Λ2 where ρ0denotes the traceless Ricci tensor, ρ0(X, Y ) = ρ(X, Y )−(τ/4) g(X, Y ) and W denotes the Weyl conformal curvature tensor given by W(X, Y, V, W) = R(X, Y, V, W) + τ (n−1)(n−2)ng(X, V )g(Y, W)−g(Y, V )g(X, W)o −1 n−2nρ(X, V )g(Y, W)−ρ(Y, V )g(X, W) +ρ(Y, W)g(X, V )−ρ(X, W)g(Y, V )o. The Hodge star operator ∗: Λ2→Λ2associated with any (2,2) metric induces a further splitting Λ2= Λ2 +⊕Λ2 −, where Λ2 ±denotes the ±1–eigenspaces of the Hodge star operator, that is, Λ2 ±={α∈Λ2(M) : ∗α=±α}. Then, the curvature tensor decomposes as R=τ 12 IdΛ2+ρ0+W++W−, where W±= (W±∗W)/2. Recall that a semi–Riemannian four–dimensional manifold is called self–dual (resp. anti–self–dual) if W−= 0 (resp. W+= 0). An interesting feature of four–dimensional Osserman metrics comes from the fact that an algebraic curvature tensor in a four–dimensional vector space is Osserman if and only if it is Einstein and self-dual for an appropriate orientation of the underlying vector space. 29 30 3 Four–dimensional Osserman metrics Therefore, pointwise Osserman metrics in dimension four are those which are Einstein and self–dual or anti–self–dual. Let {e1, e2, e3, e4}be an orthonormal basis with e1and e2spacelike vectors and e3and e4timelike vectors. Local bases of the spaces of self–dual and anti–self–dual two–forms may be constructed as Λ2 ±= span ©E± 1, E± 2, E± 3ª, where E± 1=e1∧e2±e3∧e4 √2, E± 2=e1∧e3±e2∧e4 √2, E± 3=e1∧e4∓e2∧e3 √2. We observe that the Hodge star operator satisfies ei∧ej∧?(ek∧el) = ¡δi kδj l−δi lδj k¢εiεje1∧e2∧e3∧e4, where εi=g(ei, ei). Note that hE± 1, E± 1i= 1, hE± 2, E± 2i=hE± 3, E± 3i=−1. Then, with respect to the above bases the self–dual and the anti–self–dual Weyl curvature operators W±: Λ2 ±→Λ2 ±have the matrix representation W±=   W± 11 W± 12 W± 13 −W± 12 −W± 22 −W± 23 −W± 13 −W± 23 −W± 33   , where W± ij =W(E± i, E± j) and W(ei∧ej, ek∧el) = W(ei, ej, ek, el). For any non–null vector xin the (2,2) setting, the induced metric on Rx⊥is of Lorentzian signature, and hence, the eigenvalue structure does not completely characterize the Jacobi operator Rx. The consideration of the Jordan normal form led to introduce the so–called Jordan–Osserman metrics (see Subsection 2.2.1). Four–dimensional Jordan– Osserman metrics were initially investigated by N. Blaˇzi´c, N. Bokan and Z. Raki´c [21] who considered four different possibilities according to the behavior of the Jordan normal form of the Jacobi operators. These four types are: (Ia) The Jacobi operator is diagonalizable, Rx=  α β γ . (Ib) The Jacobi operator has a complex eigenvalue, Rx=  α−β β α γ . (II) The minimal polynomial of the Jacobi operator has a double root, Rx=  α β 1β . 3.1 New examples of Osserman metrics with non–diagonalizable Jacobi operators 31 (III) The minimal polynomial of the Jacobi operator has a triple root, Rx=  α 1α 1α . Moreover, there is a one to one correspondence between the different possibilities of the Jacobi operators of Types Ia, Ib, II, III and the Jordan normal form of the (anti–)self–dual part of the Weyl conformal curvature tensor [62]. It has been shown in [21] that four–dimensional Osserman metrics with diagonalizable Jacobi operators are locally isometric to a real, complex or para–complex space form and that Type Ib metrics cannot occur. Moreover, a locally symmetric Osserman (2,2) metric has diagonalizable Jacobi operators or it is isometric with some Type II metric to nilpotent Jacobi operators [65]. The fact that all known examples of non–symmetric Osserman metrics had 2–step or 3–step nilpotent Jacobi operators suggested that no other examples exist [62], [68]. This was conjectured by several authors. Our purpose is to show the existence of such metrics (Section 3.1) and to give a complete description of them (Sections 3.2 and 3.3). 3.1 New examples of Osserman metrics with non– diagonalizable Jacobi operators Let us take the usual coordinates (x1, x2, x3, x4) in M=R4. For any arbitrary real–valued function fand any non–zero constant kwe define the metric [48] g=dx1⊗dx3+dx3⊗dx1+dx2⊗dx4+dx4⊗dx2+µ4kx2 1−1 4kf(x4)2¶dx3⊗dx3 +4kx2 2dx4⊗dx4+µ4kx1x2+x2f(x4)−1 4kf0(x4)¶(dx3⊗dx4+dx4⊗dx3). The following two lemmas can be obtained after some tedious but straightforward calculations from the definition of the metric g. Lemma 3.1. The Christoffel symbols associated with gare Γ1 13 =−Γ3 33 = 4kx1,Γ2 13 = Γ1 14 =−Γ3 34 =1 2Γ2 24 =−1 2Γ4 44 = 2kx2, Γ2 23 = Γ1 24 =−Γ4 34 =1 2¡4kx1+f(x4)¢,Γ1 33 = 16k2x3 1−x1f(x4)2, Γ2 33 =x1¡16k2x1x2−f0(x4)¢+f(x4)µ4kx1x2+f0(x4) 4k¶, Γ1 34 = 16k2x2 1x2+ 4kx1x2f(x4)−1 2x1f0(x4)−3f(x4)f0(x4) 8k, Γ2 34 =1 2x2¡32k2x1x2+ 8kx2f(x4)−f0(x4)¢, Γ1 44 = 16k2x1x2 2+ 4kx2 2f(x4)−f00(x4) 4k,Γ2 44 = 16k2x3 2. 32 3 Four–dimensional Osserman metrics From the previous lemma we get Lemma 3.2. The curvature tensor of gis determined by R1313 =R2424 =−4k, R1324 =R1423 =−2k, R1334 =kx2¡4kx1+f(x4)¢, R1434 = 4k2x2 2, R2334 =f(x4)2 4−4k2x2 1, R2434 =f0(x4) 2−kx2¡4kx1+f(x4)¢, R3434 =f0(x4)2 4k+ 2kx1x2f0(x4)−2kx2 2f(x4)2−x1f00(x4) −f(x4)µ8k2x1x2 2−5 2x2f0(x4)−f00(x4) 4k¶. Now, we calculate the Jacobi operator associated with g. We have the following theorem [48]. Theorem 3.3. For any function f, the metric gis Osserman of signature (2,2) with eigenvalues {0,4k, k, k}. Moreover, the Jacobi operators are diagonalizable if and only if 24kf(x4)f0(x4)x2−12kf00(x4)x1+ 3f(x4)f00(x4)+4f0(x4)2= 0. Otherwise, kis a double root of the minimal polynomial of the Jacobi operators and (M, g) is Jordan–Osserman on the open set where the above equation does not hold. Proof. The eigenvalues of the Jacobi operator of an Osserman metric change sign when passing from timelike to spacelike directions. Thus, for the purpose of studying the Osserman property, it is convenient to consider the normalized Jacobi operator JR(X) = g(X, X)−1RXassociated with each non–null vector X, whose eigenvalues are constant if and only if (M, g) is Osserman. Let X=P4 i=1 αi∂ibe a non–null vector, where {∂i=∂/∂xi} denotes the coordinate basis. The associated Jacobi operator RX=R(X, ·)Xcan be expressed with respect to the coordinate basis {∂i}as (3.1) RX=      a11 a12 a13 a14 a21 a22 a23 a24 −4kα2 3−4kα3α4a33 a34 −4kα3α4−4kα2 4a43 a44      , with a11 = 5kx2f(x4)α3α4−f(x4)2α2 3+ 2k¡2α1α3+α2α4 +2k(4x2 1α2 3+ 5x1x2α3α4+x2 2α2 4)¢−α3α4f0(x4), a12 =1 4α4¡12kx2f(x4)α4−3f(x4)2α3+ 8k(α1+ 6kx1(x1α3+x2α4)) −2α4f0(x4)¢, 3.1 New examples of Osserman metrics with non–diagonalizable Jacobi operators 33 a13 =1 16k¡f(x4)2α3(16kα1+α4f0(x4)) + 4f(x4)α4(7kx2α4f0(x4)−16k2x2α1 +α4f00(x4)) −2(4kα4(2kx1(x1α3+x2α4)−α1)f0(x4) −3α2 4f0(x4)2+ 8k(4kα1(α1+ 4kx1(x1α3+x2α4)) + x1α2 4f00(x4)))¢, a14 =−1 16k¡f(x4)2α3(α3f0(x4)−12kα2)+4f(x4)(4k2x2(α1α3+ 3α2α4) +7kx2α3α4f0(x4) + α3α4f00(x4)) + 2(−4k(α1α3+α2α4 +2kx1α3(x1α3+x2α4))f0(x4) + 3α3α4f0(x4)2+ 8k(4k(3kx1α2(x1α3+x2α4) +α1(α2+kx2(x1α3+x2α4))) −x1α3α4f00(x4)))¢, a21 =α3¡3kx2f(x4)α3+ 2k(α2+ 6kx2(x1α3+x2α4)) −α3f0(x4)¢, a22 = 2kα1α3−1 4f(x4)2α2 3+ 4k2x2 1α2 3+ 4kα2α4+ 5kf(x4)x2α3α4 +20k2x1x2α3α4+ 16k2x2 2α2 4−3 2α3α4f0(x4), a23 =−1 4k¡4k(kx2α4(x1α3+x2α4)−α1α3)f0(x4)−kf(x4)2α2α3+α3α4f0(x4)2 +f(x4)(4k2x2(3α1α3+α2α4) + 9kx2α3α4f0(x4) + α3α4f00(x4)) +4k(4k(kx1α2(x1α3+x2α4) + α1(α2+ 3kx2(x1α3+x2α4))) −x1α3α4f00(x4)), a24 =1 4k¡2kα3(3α2+ 2kx2(x1α3+x2α4))f0(x4) + α2 3f0(x4)2+f(x4)α3(−16k2x2α2 +9kx2α3f0(x4) + α3f00(x4)) −4k(4kα2(α2+ 4kx2(x1α3+x2α4)) + x1α2 3f00(x4))¢, a33 =k¡4α1α3+x2f(x4)α3α4+ 2α4(α2+ 2kx2(x1α3+x2α4))¢, a34 =−kα3¡−2α2+x2f(x4)α3+ 4kx2(x1α3+x2α4)¢, a43 =1 4α4¡f(x4)2α3−4kx2f(x4)α4+ 2(4k(α1−2kx1(x1α3+x2α4)) + α4f0(x4))¢, a44 =kx2f(x4)α3α4−1 4f(x4)2α2 3+ 2k(α1α3+ 2(α2α4+kx1α3(x1α3+x2α4))) −1 2α3α4f0(x4). Using the above expressions we get that the characteristic polynomial of JR(X) is given by pJR(X)(λ) = λ(λ−4k)(λ−k)2, and thus the metric gis Osserman with eigenvalues {0,4k, k, k}. In order to analyze the diagonalizability of the Jacobi operators, we consider the minimal polynomials mJR(X)(λ). It follows after some calculations that JR(X)·(JR(X)−4kId)·(JR(X)−kId) = k 4g(X, X)−1Ξ    0 0 −α2 4α3α4 0 0 α3α4−α2 3 0 0 0 0 0 0 0 0    , 40 3 Four–dimensional Osserman metrics neither diagonalizable nor nilpotent. This motivates the study of a more general situation: Walker four–dimensional manifolds equipped with a para–Hermitian structure, which we tackle in the following section. 3.2.1 Einstein para-Hermitian structures on Walker manifolds Let gbe a Walker metric expressed in the above coordinates (x1, . . . , x4). There is a natural almost para–Hermitian structure Jdefined by J∂1=−∂1, J∂2=∂2, J∂3=−a∂1+∂3, J∂4=b∂2−∂4, where as usual, ∂i=∂/∂xi. Throughout this section we use subscripts to denote partial derivatives of functions, that is, for each function hdepending on (x1, . . . , x4) we write hi=∂h/∂xi. After doing some straightforward calculations we determine the Levi–Civita connection of a Walker metric. Lemma 3.12. The non–vanishing components of the Levi–Civita connection are ∇∂1∂3=1 2a1∂1+1 2c1∂2,∇∂1∂4=1 2c1∂1+1 2b1∂2, ∇∂2∂3=1 2a2∂1+1 2c2∂2,∇∂2∂4=1 2c2∂1+1 2b2∂2, ∇∂3∂3=1 2(aa1+ca2+a3)∂1+1 2(ca1+ba2−a4+ 2c3)∂2−a1 2∂3−a2 2∂4, ∇∂3∂4=1 2(a4+ac1+cc2)∂1+1 2(b3+cc1+bc2)∂2−c1 2∂3−c2 2∂4, ∇∂4∂4=1 2(ab1+cb2−b3+ 2c4)∂1+1 2(cb1+bb2+b4)∂2−b1 2∂3−b2 2∂4. By analyzing the almost para–Hermitian structure Jwe obtain the following Theorem 3.13. The Walker metric gequipped with the almost para–Hermitian structure J is para–Hermitian if and only if a2=b1= 0. Moreover, the almost para–K¨ahler condition holds if and only if c1=c2= 0 and hence the para–K¨ahler condition is equivalent to a2=b1=c1=c2= 0. Proof. We write J∂i=PjJj i∂j. The components of the Nijenhuis tensor are determined by Ni jk = 2 4 X l=1 ÃJl j ∂Ji k ∂xl−Jl k ∂Ji j ∂xl−Ji l ∂Jl k ∂xj +Jl j ∂Jl j ∂xk!. The non–zero components are N2 14 = 4b1, N1 23 = 4a2, N1 43 =−2ba2, N2 43 = 2ab1. 3.2.1 Einstein para-Hermitian structures on Walker manifolds 41 Hence, the integrability of Jis characterized by a2=b1= 0. On the other hand, the second part of the result is obtained after a direct and straightforward calculation from Lemma 3.12. In the rest of this section we study four–dimensional Walker metrics equipped with the para–Hermitian structure J. We obtain a classification of Einstein para–Hermitian Walker metrics as a first step to analyze the Osserman condition for Walker manifolds. Using Lemma 3.12, we calculate the Riemannian curvature tensor after some tedious calculations: Lemma 3.14. The curvature tensor of the Walker metric gis given by R1313 =−1 2a11, R1314 =−1 2c11, R1323 =−1 2a12, R1324 =−1 2c12, R1414 =−1 2b11, R1423 =−1 2c12, R1424 =−1 2b12, R2424 =−1 2b22, R2323 =−1 2a22, R2324 =−1 2c22, R2434 =1 4(a2b1−c1c2−2b23 + 2c24), R1334 =1 4(−a2b1+c1c2+ 2a14 −2c13), R1434 =1 4¡−c2 1+a1b1−b1c2+b2c1−2b13 + 2c14¢, R2334 =1 4¡c2 2−a2b2−a1c2+a2c1+ 2a24 −2c23¢, R3434 =1 4¡−ac2 1−bc2 2+aa1b1+ca1b2−a1b3+ 2a1c4+ca2b1+ba2b2+a2b4 +a3b1−a4b2−2a4c1+ 2b2c3−2b3c2−2cc1c2−2a44 −2b33 + 4c34¢. Using the previous result we calculate the Ricci tensor and the scalar curvature. Lemma 3.15. The Ricci tensor of the four–dimensional Walker metric gis given by ρ13 =1 2(a11 +c12), ρ14 =1 2(b12 +c11), ρ23 =1 2(a12 +c22), ρ24 =1 2(b22 +c12), ρ33 =1 2¡−c2 2+a1c2+a2b2−a2c1+aa11 + 2ca12 +ba22 + 2c23 −2a24¢, ρ34 =1 2(−a2b1+c1c2+a14 +b23 +ac11 + 2cc12 −c13 +bc22 −c24), ρ44 =1 2¡−c2 1+a1b1−b1c2+b2c1+ab11 + 2cb12 −2b13 +bb22 + 2c14¢. As a consequence, the scalar curvature is τ=a11 +b22 + 2c12. 42 3 Four–dimensional Osserman metrics Corollary 3.16. The traceless Ricci tensor ρ0=ρ−(τ/4)gof the Walker metric gis determined by ρ0 13 =−ρ0 24 =1 4(a11 −b22), ρ0 14 =1 2(b12 +c11), ρ0 23 =1 2(a12 +c22), ρ0 33 =1 4¡2a1c2+ 2a2b2−2a2c1−2c2 2+a(a11−b22)+4ca12 + 2ba22 −4a24 −2ac12 + 4c23¢, ρ0 44 =1 4¡2a1b1−2b1c2+ 2b2c1−2c2 1−b(a11 −b22)+2ab11 + 4cb12 −4b13 −2bc12 + 4c14¢, ρ0 34 =1 4(−2a2b1+ 2c1c2−c(a11 −2c12 +b22)+2a14 + 2b23 + 2ac11 −2c13 + 2bc22 −2c24). We are now ready to characterize Einstein para–Hermitian Walker metrics. Theorem 3.17. The four dimensional Walker metric gequipped with the almost para– Hermitian structure Jis Einstein para-Hermitian if and only if the defining functions a,b and care any of the following types: (A) The scalar curvature τvanishes and a,band ccan be written as a=a(x1, x3, x4) = x1P(x3, x4) + ξ(x3, x4), b=b(x2, x3, x4) = x2Q(x3, x4) + η(x3, x4), c=c(x1, x2, x3, x4) = x1S(x3, x4) + x2T(x3, x4) + γ(x3, x4), where ξ,ηand γare arbitrary smooth functions, and P,Q,S,Tare smooth functions satisfying PT −T2+ 2T3= 0, QS −S2+ 2S4= 0, ST +Q3−S3+P4−T4= 0. (B) The scalar curvature τis non–zero and a,band csatisfy a=a(x1, x3, x4) = τ 4x2 1+x1P(x3, x4) + ξ(x3, x4), b=b(x2, x3, x4) = τ 4x2 2+x2Q(x3, x4) + η(x3, x4), c=c(x3, x4) = 2 τ(P4(x3, x4) + Q3(x3, x4)) , where P,Q,ξand ηare arbitrary smooth functions. (C) The scalar curvature τis non–zero and a,band ccan be written as a=a(x1, x3, x4) = τ 6x2 1+x1P+6 τ¡PT −T2+ 2T3¢, b=b(x2, x3, x4) = τ 6x2 2+x2Q+6 τ¡QS −S2+ 2S4¢, c=c(x1, x2, x3, x4) = τ 6x1x2+x1S+x2T+6 τ(ST +Q3−S3+P4−T4), for any smooth functions P,Q,Sand Tdepending on (x3, x4). 3.2.1 Einstein para-Hermitian structures on Walker manifolds 43 Proof. Since Jis para–Hermitian, we have a2=b1= 0 by Theorem 3.13. Hence, a= a(x1, x3, x4) and b=b(x2, x3, x4). Since gis Einstein, we have ρ0= 0. Using the previous fact for the functions aand band Corollary 3.16 we get (3.2) a11 −b22 =c11 =c22 = 0, 2a1c2−2c2 2−2ac12 + 4c23 = 2b2c1−2c2 1−2bc12 + 4c14 = 0, 2c1c2−ca11 + 2a14 −cb22 + 2b23 + 2cc12 −2c13 −2c24 = 0. We separate the proof of this theorem in three steps. Claim 3.18. The functions a,band cdefining the metric gsatisfy a=a(x1, x3, x4) = x2 1κ(x3, x4) + x1P(x3, x4) + ξ(x3, x4), b=b(x2, x3, x4) = x2 2κ(x3, x4) + x2Q(x3, x4) + η(x3, x4), c=c(x1, x2, x3, x4) = x1x2α(x3, x4) + x1S(x3, x4) + x2T(x3, x4) + γ(x3, x4) where κ(x3, x4),P(x3, x4),Q(x3, x4),ξ(x3, x4),η(x3, x4),α(x3, x4),S(x3, x4),T(x3, x4) and γ(x3, x4)are arbitrary functions. The first equation in (3.2) and a2=b1= 0 implies a111 =b222 = 0 and hence a(resp. b) is a quadratic function of x1(resp. x2) with parameters x3and x4. Then, we can express aand bas stated in the first two equations of Claim 3.18. On the other hand, the last two equalities of the first equation in (3.2) imply that cis a linear function with respect to x1 and x2, taking the form of the third equation of Claim 3.18. Claim 3.19. The functions a,band ccan be written as a=a(x1, x3, x4) = κ x2 1+x1P(x3, x4) + ξ(x3, x4), b=b(x2, x3, x4) = κ x2 2+x2Q(x3, x4) + η(x3, x4), c=c(x1, x2, x3, x4) = ³τ 2−2κ´x1x2+x1S(x3, x4) + x2T(x3, x4) + γ(x3, x4), where κis a constant and P(x3, x4),Q(x3, x4),ξ(x3, x4),η(x3, x4),S(x3, x4),T(x3, x4) and γ(x3, x4)are arbitrary functions. Moreover, one of the following three possibilities can occur: κ=τ= 0 or, in case τ6= 0, either κ=τ 4or κ=τ 6. Lemma 3.15 combined with Claim 3.18 implies τ= 4κ(x3, x4)+2α(x3, x4). Hence, α(x3, x4) = τ/2−2κ(x3, x4). We recall that, since gis Einstein, the scalar curvature τis constant. Differentiating the second equation in (3.2) twice with respect to x1, we get τ2−10τκ(x3, x4) + 24κ(x3, x4)2= 0. Thus, κ(x3, x4) must be constant and Claim 3.19 follows. We are now ready to finish the proof of Theorem 3.17. We analyze the three different possibilities which arise in Claim 3.19 separately. 44 3 Four–dimensional Osserman metrics Assume κ=τ= 0. This is the simplest case, because the expression in Claim 3.19 reduces to a=a(x1, x3, x4) = x1P(x3, x4) + ξ(x3, x4), b=b(x2, x3, x4) = x2Q(x3, x4) + η(x3, x4), c=c(x1, x2, x3, x4) = x1S(x3, x4) + x2T(x3, x4) + γ(x3, x4). Furthermore, for such functions the last two equations in (3.2) transform into PT −T2+ 2T3= 0, QS −S2+ 2S4= 0, ST +Q3−S3+P4−T4= 0, which is exactly case (A) of Theorem 3.17. Assume κ=τ/46= 0. In this case, the expression of Claim 3.19 transforms into a=a(x1, x3, x4) = τ 4x2 1+x1P(x3, x4) + ξ(x3, x4), b=b(x2, x3, x4) = τ 4x2 2+x2Q(x3, x4) + η(x3, x4), c=c(x1, x2, x3, x4) = x1S(x3, x4) + x2T(x3, x4) + γ(x3, x4). The second equation in (3.2) reduces to ¡τx1+ 2P(x3, x4)¢T(x3, x4)−2T(x3, x4)2+ 4T3(x3, x4) = 0, ¡τx2+ 2Q(x3, x4)¢S(x3, x4)−2S(x3, x4)2+ 4S4(x3, x4) = 0, which hold if and only if T(x3, x4) = S(x3, x4) = 0. Using this condition, the last equation in (3.2) leads to τγ(x3, x4)−2(P4(x3, x4)+ Q3(x3, x4)) = 0 and therefore we can determine γby γ(x3, x4) = 2 τ(P4(x3, x4) + Q3(x3, x4)). Altogether this implies case (B) of Theorem 3.17. Assume κ=τ/66= 0. In this case, the expression in Claim 3.19 yields a=a(x1, x3, x4) = τ 6x2 1+x1P(x3, x4) + ξ(x3, x4), b=b(x2, x3, x4) = τ 6x2 2+x2Q(x3, x4) + η(x3, x4), c=c(x1, x2, x3, x4) = τ 6x1x2+x1S(x3, x4) + x2T(x3, x4) + γ(x3, x4), and a straightforward calculation shows that the last two equations in (3.2) transform into τ 6ξ(x3, x4)−(P(x3, x4)T(x3, x4)−T(x3, x4)2+ 2T3(x3, x4)) = 0, τ 6η(x3, x4)−(Q(x3, x4)S(x3, x4)−S(x3, x4)2+ 2S4(x3, x4)) = 0, τ 6γ(x3, x4)−(S(x3, x4)T(x3, x4) + Q3(x3, x4)−S3(x3, x4) + P4(x3, x4)−T4(x3, x4)) = 0, 3.2.2 Osserman para–Hermitian structures on Walker manifolds 45 from where we can determine ξ(x3, x4), η(x3, x4) and γ(x3, x4) as follows ξ(x3, x4) = 6 τ(P(x3, x4)T(x3, x4)−T(x3, x4)2+ 2T3(x3, x4)), η(x3, x4) = 6 τ(Q(x3, x4)S(x3, x4)−S(x3, x4)2+ 2S4(x3, x4)), γ(x3, x4) = 6 τ(S(x3, x4)T(x3, x4) + Q3(x3, x4)−S3(x3, x4) + P4(x3, x4)−T4(x3, x4)). Altogether this implies case (C) in Theorem 3.17, which finishes the proof. 3.2.2 Osserman para–Hermitian structures on Walker manifolds In this section we analyze the Osserman condition for the three families of Einstein para– Hermitian Walker metrics determined in Theorem 3.17. We study each case separately. Einstein para–Hermitian metrics of type (A) Einstein para–Hermitian Walker metrics of type (A) defined in Theorem 3.17 are Osserman, but they do not provide the new desired examples. Indeed, if X=P4 i=1 αi∂iis an arbitrary vector then the associated Jacobi operator, when expressed in the coordinate basis, has the form RX=ÃA B 0t A!,where A=Ψ 4Ã−α3α4−α2 4 α2 3α3α4! and Ψ = Q3+S3−P4−T4. Hence the characteristic polynomial of the Jacobi operators is pRX(λ) = λ4(independently of the 2 ×2-matrix B). Therefore the Jacobi operators are either vanishing or nilpotent. Einstein para–Hermitian metrics of type (B) Metrics in the family (B) of Theorem 3.17 are not Osserman. To see this, recall that a four–dimensional semi–Riemannian manifold is pointwise Osserman if and only if there is a choice of orientation such that the manifold is Einstein self–dual (or anti–self–dual). See [3], [62]. Given the Walker metric g, we have that e1=1 2(1 −a)∂1+∂3, e2=−c∂1+1 2(1 −b)∂2+∂4, e3=−1 2(1 + a)∂1+∂3, e4=−c∂1−1 2(1 + b)∂2+∂4 defines an orthonormal basis of the tangent space. Local bases of the spaces of self–dual and anti–self–dual two–forms can be constructed as Λ2 ±= span ©E± 1, E± 2, E± 3ª, where E± 1=e1∧e2±e3∧e4 √2, E± 2=e1∧e3±e2∧e4 √2, E± 3=e1∧e4∓e2∧e3 √2. 46 3 Four–dimensional Osserman metrics A long but direct and straightforward calculation using Lemmas 3.14 and 3.15 and the definition of the Weyl tensor shows that W+ 22 =W− 22 =−τ/6, and hence para–Hermitian Walker metrics of type (B) cannot be Osserman. See Lemma 3.23 for details. Einstein para–Hermitian metrics of type (C) This last family of Einstein para–Hermitian Walker metrics will provide the desired examples of Osserman spaces. In particular, we have the following Theorem 3.20. An Einstein para–Hermitian Walker metric of type (C) is Osserman of signature (2,2) with eigenvalues {0, τ/6, τ/24, τ/24}. Proof. After a long but straightforward calculation one gets that (see Lemma 3.23) W−= 0, W+=   W+ 11 W+ 12 W+ 11 +τ 12 −W+ 12 τ 6−W+ 12 −(W+ 11 +τ 12)−W+ 12 −(W+ 11 +τ 6)   , and hence it follows that W+has eigenvalues {τ/6,−τ/12,−τ/12}. As a consequence, any Einstein para–Hermitian Walker metric defined by Theorem 3.17 (C) is Osserman (Einstein self–dual) and thus the eigenvalues of the self–dual operator W+determine the eigenvalues of the Jacobi operators, which turn out to be {0, τ/6, τ/24, τ/24}. Remark 3.21.Note that the metric studied in Section 3.1 is a particular case of the general family of Einstein para–Hermitian Walker metrics of Theorem 3.17 (C). 3.3 General description of Osserman metrics whose Jacobi operators have two distinct non-zero eigenvalues The purpose of this section is to clarify the situation of Type II Jordan–Osserman metrics by proving the following [49] Theorem 3.22. Let (M, g)be a four–dimensional Type II Jordan–Osserman manifold. Then the Jacobi operators are either two–step nilpotent or there exist local coordinates (x1, . . . , x4)such that the metric is given by dx1⊗dx3+dx3⊗dx1+dx2⊗dx4+dx4⊗dx2+ 4 X i,j=3 sij dxi⊗dxj 3.3.1 Self-duality and anti-self-duality conditions 47 for some functions sij(x1, . . . , x4)which can be written as s33 =x2 1 τ 6+x1P+x2Q+6 τ©Q(T−U) + V(P−V)−2(Q4−V3)ª, s44 =x2 2 τ 6+x1S+x2T+6 τ©S(P−V) + U(T−U)−2(S3−U4)ª, s34 =s43 =x1x2 τ 6+x1U+x2V+6 τ©−QS +UV +T3−U3+P4−V4ª, where P,Q,S,T,Uand Vare arbitrary functions depending on the coordinates (x3, x4). The proof of Theorem 3.22 is based on the following facts: 1. A four–dimensional semi–Riemannian manifold is pointwise Osserman if and only if it is Einstein self–dual (or anti–self–dual) [3], [80]. 2. A Type II Jordan–Osserman metric is either Ricci flat (that is, α=β= 0) or β= 4α6= 0 [21, Corollary 8.3]. 3. A Type II Jordan–Osserman metric whose Jacobi operators are not nilpotent (that is, α= 4β6= 0) admits a local parallel field of two–dimensional planes [21, Proposition 8.4]. Therefore, we investigate Walker metrics (which are those admitting a locally defined two–dimensional degenerate parallel distribution) in detail in Subsection 3.3.1, with special attention to the (anti–)self–dual Weyl curvature tensors. A complete description of self– dual Walker metrics is given in Subsection 3.3.2. The integration of the Einstein equation for a self–dual Walker metric, which lets us determine all pointwise Osserman self–dual Walker metrics, is carried out in Subsection 3.3.3. This leads to the proof of Theorem 3.22. 3.3.1 Self-duality and anti-self-duality conditions In this section we obtain the expression of the self–dual and the anti–self–dual Weyl conformal curvature tensors for the Walker metric ggiven at the beginning of Section 3.2 with respect to an orthonormal basis {e1, . . . , e4}where e1=1 2(1 −a)∂1+∂3, e2=−c∂1+1 2(1 −b)∂2+∂4, e3=−1 2(1 + a)∂1+∂3, e4=−c∂1−1 2(1 + b)∂2+∂4. A long but straightforward calculation using Lemma 3.14 and the expressions for the Ricci tensor and the scalar curvature in Lemma 3.15 implies the following lemma. 48 3 Four–dimensional Osserman metrics Lemma 3.23. With respect to the above basis, the components of W−are given by W− 11 =−1 12(a11 + 3a22 + 3b11 +b22 −4c12), W− 22 =−1 6(a11 +b22 −4c12), W− 33 =1 12(a11 −3a22 −3b11 +b22 −4c12), W− 12 =1 4(a12 +b12 −c11 −c22), W− 13 =1 4(a22 −b11), W− 23 =−1 4(a12 −b12 +c11 −c22). The components of W+are determined by W+ 11,W+ 12 and the scalar curvature as follows W+ 22 =−τ 6, W+ 33 =W+ 11 +τ 6, W+ 13 =W+ 11 +τ 12, W+ 23 =W+ 12. Finally, we have the expressions for W+ 11 and W+ 12: W+ 11 =1 12³6ca1b2−6a1b3−6ba1c2+ 12a1c4−6ca2b1+ 6a2b4+ 6ba2c1+ 6a3b1−6a4b2 −12a4c1+ 6ab1c2−6ab2c1+ 12b2c3−12b3c2−a11 −12c2a11 −12bca12 + 24ca14 −3b2a22 + 12ba24 −12a44 −3a2b11 + 12ab13 −b22 −12b33 + 12acc11 −2c12 + 6abc12 −24cc13 −12ac14 −12bc23 + 24c34´, W+ 12 =1 4(−2ca11 −ba12 + 2a14 +ab12 −2b23 +ac11 −2cc12 −2c13 −bc22 + 2c24). Remark 3.24.The connection between Einstein (anti–)self–dual and pointwise Osserman manifolds goes further to the Jordan normal forms of the non–zero part of the Weyl curvature tensor W±and the Jacobi operators (see [62]). Pointwise Osserman manifolds whose Jacobi operators are of Type Ia, Ib, II or III correspond to self–dual (or anti–self–dual) Einstein manifolds whose self–dual (or anti–self–dual) Weyl curvature tensor is of Type Ia, Ib, II or III, respectively. Lemma 3.23 shows that W+=   W+ 11 W+ 12 W+ 11 +τ 12 −W+ 12 τ 6−W+ 12 −(W+ 11 +τ 12)−W+ 12 −(W+ 11 +τ 6)   , and, as a consequence, the eigenvalues of W+are {τ/6,−τ/12,−τ/12}. Since the induced metric on Λ2 +has Lorentzian signature, the structure of W+is determined by its Jordan normal form, which may correspond to Type Ia or Type II/III, depending on whether W+ is diagonalizable or not. A straightforward calculation shows that ³W+−τ 6Id´·³W++τ 12Id´=τ2+ 12τW+ 11 + 48 ¡W+ 12¢2 48  −1 0 −1 0 0 0 1 0 1  , from where we have the following: 3.3.2 Explicit form of self-dual Walker metrics 49 (i) If τ6= 0, we have that W+has non–zero eigenvalues {τ/6,−τ/12,−τ/12}and the equality τ2+ 12τW+ 11 + 48 ¡W+ 12¢2= 0 is the necessary and sufficient condition for the diagonalizability of W+. If the above equation does not hold, then −τ/12 is a double root of the minimal polynomial of W+. (ii) If τ= 0, then W+vanishes if and only if W+ 11 =W+ 12 = 0 and moreover 1. W+is two–step nilpotent if and only if W+ 11 6= 0 and W+ 12 = 0, 2. W+is three–step nilpotent if and only if W+ 12 6= 0. On the other hand, taking into account the eigenvalues of W+, any anti–self–dual Walker metric has vanishing scalar curvature and hence Einstein anti–self–dual Walker metrics are Ricci flat. 3.3.2 Explicit form of self-dual Walker metrics Our main purpose is to obtain a description of non–Ricci flat Type II Jordan–Osserman four–dimensional manifolds. As a consequence of Remark 3.24 we may restrict our analysis to self–dual Walker metrics. In this section we give a complete description of self–dual Walker metrics by integrating the partial differential equations obtained from Lemma 3.23. Theorem 3.25. A Walker metric gis self–dual if and only if the defining functions a,b and care given by a=x3 1A+x2 1B+x2 1x2C+x1x2D+x1P+x2Q+ξ, b=x3 2C+x2 2E+x1x2 2A+x1x2F+x1S+x2T+η, c=1 2x2 1F+1 2x2 2D+x2 1x2A+x1x2 2C+1 2x1x2(B+E) + x1U+x2V+γ, where P,Q,S,T,U,V,A,B,C,D,E,F,ξ,ηand γare functions depending on the coordinates (x3, x4). Proof. Using Lemma 3.23 the self–duality can be initially characterized by means of the following five equations (3.3) a22 =b11 =a12 −c22 =b12 −c11 =a11 +b22 −4c12 = 0. Claim 3.26. We have a(x1, x2, x3, x4) = x2A(x1, x3, x4) + B(x1, x3, x4), b(x1, x2, x3, x4) = x1C(x2, x3, x4) + D(x2, x3, x4), c(x1, x2, x3, x4) = 1 2x2 2A1(x1, x3, x4) + x2E(x1, x3, x4) + F(x1, x3, x4). for differentiable functions A,B,C,D,Eand F. 56 Part II Curvature invariants of geodesic spheres and geodesic celestial spheres 57 In order to study the geometry of a Riemannian manifold (M, g) it is often useful to consider objects naturally associated with the metric structure of (M, g). These can be special hypersurfaces such as small geodesic spheres and tubes, bundles with (M, g) as base manifold or families of transformations reflecting symmetry properties of (M, g) [128]. In this part of the thesis we focus on the study of geodesic spheres and their curvature in relation to the curvature of the ambient manifold. Indeed, the existence of a relationship between the curvature of a Riemannian manifold and the volume of its geodesic spheres and tubes led some authors to state the following question: “To what extent is the curvature or the geometry of a given Riemannian manifold influenced, or even determined, by the properties of certain naturally defined families of geometric objects in M?”. This problem seems very difficult to handle in such a generality. However, when one looks at manifolds with a high degree of symmetry (for example two–point homogeneous spaces), these geometric objects have nice properties and one may expect to obtain characterizations of those spaces by means of such properties. By comparing a Riemannian manifold with a model space such as a two–point homogenous space we get an idea of its geometry. Thus, by understanding the geometry of spaces with a high degree of symmetry and why their properties are characteristic of them, we get a better insight into the geometry of a Riemannian manifold. Since geodesic spheres are compact submanifolds, it makes sense to calculate their volume. A. Gray and L. Vanhecke calculated the first terms in the power series expansion of the volume of geodesic spheres [83]. They conjectured that the volume of geodesic spheres can be used to characterize Euclidean geometry. More specifically, if each geodesic sphere of a Riemannian manifold has the same volume as a Euclidean sphere of the same radius, then the manifold is flat. Although the answer is known to be affirmative in several special cases, the problem remains open in full generality. Further work on geodesic spheres involved the investigation of their geometric properties and how they influence the geometry of the ambient manifold. Certain types of manifolds can be characterized by properties of geodesic spheres [33]. In this work, B.–Y. Chen and L. Vanhecke study intrinsic and extrinsic curvatures of geodesic spheres. It turns out 59 that in many cases curvature properties provide a better understanding of geometry than volume properties. The fact that the curvature tensor of a manifold is very difficult to handle motivated the study of several kinds of simplifications of this object. We are specially interested in the so–called scalar curvature invariants. Apart from their ubiquity in Riemannian geometry, specially when studying geodesic spheres and related objects, they are of interest by themselves. See for example [113] where a nice characterization of homogeneous spaces using scalar curvature invariants is given. Our aim in Chapter 4 is to investigate curvature invariants of geodesic spheres. By integrating a scalar curvature invariant along every geodesic sphere of a manifold we get a good interplay between curvature and volume–like properties. The volume conjecture of A. Gray and L. Vanhecke may be generalized to these new objects. We see in Subsection 4.2.3 that in certain cases two–point homogeneous spaces can be characterized by the integrals of scalar curvature invariants of geodesic spheres. We emphasize that it suffices one single curvature invariant to characterize these model spaces. See Subsection 4.3.1 for examples of such curvature invariants. In addition to geodesic spheres, other objects may be considered in Riemannian geometry which are also related to the Riemannian distance function: tubes around submanifolds and disks. The former are studied in Chapter 4 and they are of interest in the last part of this thesis. Geodesic disks are the main concern of Subsection 4.3.2. They were previously investigated by O. Kowalski and L. Vanhecke with special attention to their volume properties [93], [94], [95]. In this subsection we are interested in the intrinsic geometry of the boundaries of these disks and we devote our attention to the study of their total scalar curvatures obtained by integrating the scalar curvature and the quadratic scalar curvature invariants along these boundaries. Our main result is that two–point homogeneous spaces are characterized by some of the total curvatures of the boundaries of geodesic disks among Riemannian manifolds with adapted holonomy. When the attention is turned from Riemannian manifolds to space–times, various difficulties emerge. An important characteristic of Riemannian manifolds is that they have a Riemannian distance function which is continuous and whose induced topology is the same as the topology of the manifold itself. Thus, several geometric objects such as geodesic spheres may be defined, at least locally, by means of this function. These objects are also Riemannian manifolds. They have nice properties, such as compactness and an acceptable behavior with respect to other constructions. When dealing with general semi–Riemannian manifolds there is no “semi–Riemannian distance” function. In fact, a distance–like function is only defined for space–times, but even in this case its properties are completely different from those in the Riemannian setting [7]. For example, the “Lorentzian distance” may not be continuous or bounded and geometric objects defined from it usually have awkward properties. Moreover, level sets of the Lorentzian distance function with respect to a given point are not compact and although some properties of those sets have been previously investigated, they do not seem to be adequate for the investigation of volume properties. 60 In Chapter 5 we consider a new family of geometric objects in Lorentzian geometry, namely geodesic celestial spheres. Roughly speaking, they are the set of points reached after a fixed distance travelling along radial geodesics emanating from a point which are orthogonal to a given timelike direction. In Relativity, a unit timelike vector represents an instantaneous observer and the vector subspace which is orthogonal to it is called the infinitesimal rest–space, that is, the infinitesimal Newtonian universe where the observer perceives particles as Newtonian particles relative to his rest position. Then, a geodesic celestial sphere is nothing but the image by the exponential map of a celestial sphere in the infinitesimal rest–space. Following the idea of characterizing spaces with high degree of symmetry by means of volume properties of geometric objects, we carry out in Section 5.2 the calculation of the volume of geodesic celestial spheres. This depends on the radius, the base point and the instantaneous observer employed to define it. Nonetheless, in an isotropic Lorentzian manifold, this measure depends only on the radius. We see that this property is characteristic of locally isotropic Lorentzian manifolds. We discuss volume comparison results and give Bishop–G¨unther and Gromov type theorems for these objects in Section 5.2. Finally, in Section 5.3 we accomplish the characterization of locally isotropic Lorentzian manifolds using integrals of scalar curvature invariants of geodesic celestial spheres in the spirit of Chapter 4. We take advantage of the results of Subsection 4.2.3 to get this characterization. In this part we tried to keep calculations to a minimum in order to make the work more readable. A package implementing the basic identities of curvature tensors has been developed by the author [39]. This package allows us to perform calculations involving scalar curvature invariants and integration along geodesic spheres. We can obtain both explicit expressions in two–point homogeneous spaces and power series expansions in general Riemannian manifolds. 61 62 Chapter 4 The Riemannian setting Let p, q ∈Mbe two points of a Riemannian manifold Mand c: [a, b]→Ma curve joining pand q, that is, c(a) = pand c(b) = q. The length of cis given by L(c) = Rb akc0(t)kdt. The Riemannian distance between the points pand qis defined as d(p, q) = inf{L(c) : cjoins pand q}. This function is indeed a distance in Mand the induced topology of dcoincides with the topology of Mas a topological manifold. We emphasize at this point that the above definition is characteristic of Riemannian geometry and cannot be directly generalized to the indefinite signature setting. Given a point m∈M, the geodesic spheres of Mcentered at mare the level sets of the Riemannian distance function with respect to m, that is, {p∈M:d(p, m) = r}for each radius r > 0. For sufficiently small radius r, these level sets are Riemannian hypersurfaces of M. Nevertheless, for some radii it might happen that these level sets have codimension greater that one or they fail to be submanifolds of M. The latter case is not of interest to us in this chapter and we restrict our definition of geodesic spheres to sufficiently small radii so that they are compact Riemannian submanifolds. A geodesic sphere can also be defined as the image by the exponential map of a Euclidean sphere of the tangent space at a point. The fact that the Riemannian metric is positive definite ensures a good interplay between the exponential map and the Riemannian distance function. This definition is more operative for our purposes and provides us a good setting to perform the calculations needed in this chapter. The following chapter takes advantage of these ideas and proposes a new family of objects in Lorentzian manifolds whose properties may be used to characterize isotropy. As it was stated before, we focus on the study of scalar curvature invariants of geodesic spheres, thus contributing to the investigation of how the curvature of geodesic spheres is related to the curvature of the ambient manifold. This chapter is organized as follows. In Section 4.1 we introduce the main concepts of Jacobi vector field theory which are used both in this part and Part III. Then, we particularize this study to geodesic spheres in Section 4.2. We also introduce in this section the concept of simple Weyl invariant. By integrating simple Weyl invariants along 63 64 4 The Riemannian setting geodesic spheres we get the so–called total curvatures of geodesic spheres. After giving some properties of these objects, we focus on the characterization of two–point homogeneous spaces in Subsection 4.2.3. Finally, Section 4.3 takes advantage of our previous work to provide examples of total curvatures of geodesic spheres and disks which may be used to characterize real, complex and quaternionic space forms. 4.1 Tubes and Jacobi vector field theory Let ¯ Mbe a Riemannian manifold of dimension nand M⊂¯ Ma Riemannian submanifold of ¯ M. For fixed r > 0, we define the set GM(r) = {exp(rξ) : ξ∈T⊥M, g(ξ, ξ) = 1}. In general GM(r) is not a Riemannian submanifold of ¯ M. If Mis a compact embedded submanifold of ¯ Mit turns out that GM(r) is a compact hypersurface of ¯ Mfor sufficiently small radius. Thus, for a sufficiently small neighborhood of any point p∈M, a tube of sufficiently small radius around that neighborhood is a hypersurface. If GM(r) is a hypersurface then we say that GM(r) is the tube of radius raround M. We follow [13]. If GM(r) is a Riemannian submanifold of ¯ Mof codimension greater than one, then GM(r) is called a focal manifold of Mat distance r. Let p∈Mand c:I→¯ Ma geodesic parametrized by arc length with c(0) = p and c0(0) ∈T⊥ pM. Let F(s, t) = cs(t) be a geodesic variation of c=c0such that γ(s) = F(s, 0) = cs(0) ∈Mfor all sand let us define ξ(s) = c0 s(0) ∈T⊥M. Let ζbe the variational vector field of F. Then ζis a solution of the initial value problem ζ00 +¯ Rc0(ζ) = 0, ζ(0) = γ0(0) ∈TpM, ζ0(0) = Sξ(0)ζ(0) + ∇⊥ ζ(0)ξ. A Jacobi vector field ζalong cverifying ζ(0) ∈Tc(0)Mand ζ0(0) −Sc0(0)ζ(0) ∈T⊥ c(0)Mis called an M–Jacobi vector field. We say that c(r) is a focal point of Malong cif there exists an M–Jacobi vector field ζalong csuch that ζ(r) = 0. A focal point arising from a Jacobi vector field ζsuch that ζ(0) = 0, ζ0(0) ∈T⊥ pMand ζ(r) = 0 is a conjugate point of pin ¯ Malong c. Assume now that GM(r) is a submanifold of ¯ M. Let ξbe a smooth curve in T⊥Mwith ξ(0) = c0(0) such that g(ξ(t), ξ(t)) = 1 for all t. Then F(s, t) = exp¡t ξ(s)¢is a smooth geodesic variation of cconsisting of geodesics intersecting Mperpendicularly. Let ζbe the corresponding M–Jacobi vector field which is the variational vector field of F. Then ζis determined by the initial values ζ(0) = γ0(0) and ζ0(0) = ξ0(0), where γ(s) = F(s, 0). For any r, the curve γr(s) = F(r, s) = exp(r ξ(s)) is a smooth curve in GM(r). Then, Tc(r)GM(r) = {ζ(r) : ζis an M–Jacobi vector field along c}. Let us denote by S(r) the shape operator of GM(r). Then it follows that S(r)c0(r)ζ(r) = ζ0(r)>. 4.1 Tubes and Jacobi vector field theory 65 If GM(r) is a tube, that is, if GM(r) is a hypersurface, its shape operator can be described in an efficient way. Let X∈Tc(0) ¯ MªRc0(0), where ªdenotes the orthogonal complement. We introduce the following notation. By BXwe denote the parallel translation of Xalong the geodesic c. We define ζXas the M–Jacobi vector field along cgiven by the following initial conditions ζX(0) = X, ζ0 X(0) = Sc0(0)X, if X∈Tc(0)M, ζX(0) = 0, ζ0 X(0) = X, if X∈T⊥ c(0)MªRc0(0). We define D(r) by D(r)BX(r) = ζX(r) for all X∈Tc(0) ¯ MªRc0(0). Then Dis a Tc(r)¯ Mª Rc0(r) endomorphism–valued tensor field along cdetermined by the following initial value problem D00 +¯ Rc0◦D= 0, D(0) = µIdTpM0 0 0 ¶, D0(0) = µSc0(0) 0 0 IdT⊥ pMªRc0(0) ¶. The endomorphism D(r) is singular if and only if c(r) is a focal point of Malong c. If this is not the case, GM(r) is a tube and its shape operator in the direction of c0(r) is given by S(r)c0(r)=D0(r)D(r)−1. Of special interest is the case when Mis just a single point. This is the main concern of the rest of this chapter. Another interesting situation occurs when Mis hypersurface. We deal with this in what follows. Let M⊂¯ Mbe a hypersurface. The next calculations are local, so we may assume that Mis an oriented submanifold and its orientation is given by a global unit normal vector field ξ. Let r > 0 and define the map Φr:M−→ ¯ M p7→ Φr(p) = exp(r ξp). We denote by ηthe vector field along Φrsuch that ηr(p) = c0 p(r), where cpis the geodesic of ¯ Mdetermined by the initial conditions cp(0) = pand c0 p(0) = ξp. The map Φris smooth and parametrizes the tube of radius raround M,GM(r). Obviously, GM(r) is an immersed submanifold of ¯ Mif and only if Φris an immersion. It may happen, nonetheless, that GM(r) is a focal manifold. The fact that GM(r) has higher codimension depends on the rank of Φr. Let ζXbe an M–Jacobi vector field. We have X=ζX(0) ∈TM and ζ0 X(0) = SX because ξhas unit length and the normal bundle of Mhas rank one. Then it follows that Φr∗X=ζX(r),¯ ∇vηr=ζ0 X(r). Thus, Φris not an immersion at p∈Mif and only if Φr(p) is a focal point of Malong the geodesic cp. The dimension of the kernel of Φr∗pis called the multiplicity of the focal 72 4 The Riemannian setting Proof. The Gauss equation may be written as e Rxyvw =Rxyvw +σxvσyw −σxwσyv in this case. Using the power series expansion along a geodesic with respect to a parallel basis, Rijkl(expm(ru)) = Rijkl(m) + r∇uRijkl(m) + (r2/2!)∇2 uuRijkl(m) + ··· and plugging the above expression and the formula of Lemma 4.4 into the Gauss equation we have e R0 ijkl(u) = δikδjl −δilδjk =R0 ijkl,e R1 ijkl(u)=0, e Rα ijkl(u) = 1 (α−2)!∇α−2 u...uRijkl(m) +1 α! α−3 X β=1 µα β+ 1¶³σβ+1 ik (u)σα−β−1 jl (u)−σβ+1 il (u)σα−β−1 jk (u)´, for all α≥2. Hence, the first part of the result follows by induction. Finally, in the last equality of the above formula there are two clearly different terms. The first one ∇α−2 u···uRijkl(m)/(α−2)! is a partial scalar curvature invariant with α+2 degrees of freedom. The second addend is another partial curvature invariant with α+ 4 degrees of freedom. The last statement then follows from Lemma 4.4. The following lemma is a technical result that will be needed in Theorem 4.7. Lemma 4.6. Let (V, h,i)be an inner product vector space of dimension n > 2and tr a total trace in the space of covariant tensors of order 4νover V. If Ris an algebraic curvature tensor on V,Sc(R)its scalar curvature and Wthe algebraic invariant defined by W= tr(R⊗ν ··· ⊗R), then tr Ãν X α=1 R0⊗···⊗ α ↓ R⊗···⊗R0!=νAW(n)Sc(R). Proof. Clearly, tr(Pν α=1 R0⊗···⊗R⊗···⊗R0) is a scalar curvature invariant of degree one, and hence it is a multiple of the scalar curvature. Write a Sc(R) = tr Ãν X α=1 R0⊗···⊗ α ↓ R⊗···⊗R0!. The above formula is true for each algebraic curvature tensor Rin V. If we take R=R0 we have a n(n−1) = ν X α=1 tr  R0⊗···⊗ α ↓ R0⊗···⊗R0  =νtr (R0⊗···⊗R0) = ν n(n−1)AW(n). Thus a=ν AW(n). We are now ready to give a power series expansion of a simple Weyl invariant in a geodesic sphere. 4.2.1 Curvature and Weyl invariants 73 Theorem 4.7. Let f Wbe a simple intrinsic Weyl invariant of degree 2ν,ν > 1, in a geodesic sphere Gm(r). We have f W(expm(ru)) = s−2ν X α=−2ν rαf Wα+2ν(u) + O¡rs−2ν+1¢, where f Wα(u)is a partial curvature invariant of degree αin the direction of usuch that the degree of freedom of each Weyl invariant involved in its construction has the same parity as α. More specifically, we have f W0(u)=(n−1)(n−2)AW(n−1), f W1(u)=0, f W2(u) = ν AW(n−1) µτ−2(n+ 1) 3ρuu¶(m), f W3(u) = ν AW(n−1) µ∇uτ−n+ 2 2∇uρuu¶(m), f W4(u) = ω4(u) + ν 2AW(n−1) µ∇2 uuτ−2(n+ 3) 5∇2 uuρuu¶(m), where ω4(u)is a simple directional curvature invariant of degree four given by ω4(u) = ½ν AW(n−1)µ−2n+ 1 45 n X a,b=1 R2 uaub +1 9ρ2 uu¶ +B1 W(n−1)µkRk2−4 n X a,b,c=1 R2 uabc +4(n+ 12) 9 n X a,b=1 R2 uaub −8 3 n X a,b=1 ρabRuaub +4 9ρ2 uu¶ +B2 W(n−1)µkρk2+n2 9 n X a,b=1 R2 uaub −2n 3 n X a,b=1 ρabRuaub −2 n X a=1 ρ2 ua +3n+ 14 9ρ2 uu−2 3τρuu¶ +B3 W(n−1)µτ−2(n+ 1) 3ρuu¶2¾(m), and B1 W,B2 Wand B3 Ware polynomials satisfying 2B1 W(n−1) + (n−2)B2 W(n−1) + (n−1)(n−2)B3 W(n−1) = µν 2¶AW(n−1). Proof. Using the notation of Lemma 4.5, we have (e R⊗···⊗ e R)i1j1k1l1···iνjνkνlν= s−2ν X α=−2ν rαÃX β1+···+βν=αe Rβ1+2 i1j1k1l1(u)··· e Rβν+2 iνjνkνlν(u) !+O¡rs−2ν+1¢. By taking traces in the above expression, the result of Lemma 4.5 and the rule to compute the degrees, imply the first statement of Theorem 4.7. 74 4 The Riemannian setting Now we turn our attention to the explicit expressions of Theorem 4.7. As e Ris the curvature tensor of the geodesic sphere Gm(r), the coefficients of its power series expansion given by Lemma 4.5 are algebraic curvature tensors in u⊥=TmMªRu. Using Lemma 4.5 and Remark 4.2 we get f W0(u) = tr ³e R0 i1j1k1l1(u)··· e R0 iνjνkνlν(u)´= tr ³R0⊗ν ··· ⊗R0´ = (n−1)(n−2)AW(n−1). Using the notation of Lemma 4.5, since e R1= 0, we have f W1(u) = tr Ãν X α=1 e R0 i1j1k1l1(u)··· e R1 iαjαkαlα(u)··· e R0 iνjνkνlν(u)!= 0. Lemmas 4.5 and 4.6 yield f W2(u) = tr Ãν X α=1 e R0 i1j1k1l1(u)··· e R2 iαjαkαlα(u)··· e R0 iνjνkνlν(u)! = tr Ãν X α=1 R0 i1j1k1l1··· e R2 iαjαkαlα(u)···R0 iνjνkνlν!=νAW(n−1)Sc(e R2), and from the definition of e R2in Lemma 4.5 we get Sc(e R2) = τ−2(n+1) 3ρuu. Hence f W2(u) = νAW(n−1) µτ−2(n+ 1) 3ρuu¶(m). Similarly, using Lemmas 4.5 and 4.6 and the fact that Sc(e R3) = ∇uτ−n+2 2∇uρuu, we obtain f W3(u) = tr Ãν X α=1 e R0 i1j1k1l1(u)··· e R3 iαjαkαlα(u)··· e R0 iνjνkνlν(u)! = tr Ãν X α=1 R0 i1j1k1l1··· e R3 iαjαkαlα(u)···R0 iνjνkνlν! =ν AW(n−1) µ∇uτ−n+ 2 2∇uρuu¶(m). Finally, using the expression of e R⊗···⊗ e Rat the beginning of this proof, we get for ν > 1, f W4(u) = tr Ãν X α=1 R0 i1j1k1l1··· e R4 iαjαkαlα(u)···R0 iνjνkνlν! + trµX α<β R0 i1j1k1l1··· e R2 iαjαkαlα(u)··· e R0 iγjγkγlγ(u)··· e R2 iβjβkβlβ(u)···R0 iνjνkνlν¶. 4.2.1 Curvature and Weyl invariants 75 For the first term of the above equality we again use Lemmas 4.5 and 4.6 to get ν AW(n−1)³−2n+ 1 45 n X a,b=1 R2 uaub +1 9ρ2 uu +1 2∇2 uuτ−n+ 3 5∇2 uuρuu´(m). Now, we briefly discuss the second term of f W4(u), which is a simple directional curvature invariant of degree 4. Using the method of Lemma 4.6 to write the second addend of f W4(u) as a linear combination of curvature invariants of degree 4 associated with e R2(see the basis of curvature invariants of degree 4 (4.1)) we get the expression for ω4(u) and the relation among the polynomials B1 W,B2 Wand B3 W. We delete the details. Remark 4.8.If ν= 1 in the previous theorem, we essentially have to deal with the scalar curvature eτ. In this case, the second addend of f W4(u) does not appear and ω4(u) = 0. Then, eτ(expm(ru)) = Ps−2 α=−2rαSc(e Rα+2)+O(rs−1). See [33] and [44] for an explicit power series expansion. Example 4.9.The coefficients B1 Wand B2 Win the expression of ω4(u) for Weyl invariants of degree 4 and 6 can be given as follows [39] WkRk2kρk2τ2 B1 W(n−1) 1 0 0 B2 W(n−1) 0 1 0 Wτ3τkρk2τkRk2ˇρhρ⊗ρ, ¯ Ri hρ, ˙ Riˇ Rˇ ¯ R B1 W(n−1) 0 0 (n−1)(n−2) 0 0 n−2 6 −3/2 B2 W(n−1) 0 (n−2)(n−1) 0 3(n−2) 2n−5 4 0 3 These coefficients can also we found in [37], [38] and [44]. Now, we derive some geometrical consequences of the expansions in Theorem 4.7. We first need the following technical result. See for example [45]. Lemma 4.10. Let (M, g)be an n–dimensional Einstein manifold. If akRk2+b n X i,j,k=1 R2 uijk +c n X i,j=1 R2 uiuj =k for some real constants a,b,c,kwith (n+ 4)b+ 3c6= 0,c6= 0 and for all unit vectors u∈TM, then (M, g)is 2–stein. Proof. We define the tensors ωxyvw = n X i,j=1 RxiyjRviwj and ηxy = n X i,j,k=1 RxijkRyijk. 76 4 The Riemannian setting For all vectors x, y ∈TmMand all constants α, β ∈R, it follows from the assumption that akRk2g2 α x+β y,α x+β y +b ηα x+β y,α x+β ygα x+β y,α x+β y +c ωα x+β y,...,α x+β y =k g2 α x+β y,α x+β y. Now, we expand the previous expression and take the coefficient of α2β2. Putting y=ei and taking the trace we obtain 2akRk2(n+ 2)g(x, x) + b³kRk2g(x, x)+(n+ 4)ηxx´ +2cÃn X i,j=1 ρijRxixj +3 2ηxx!= 2(n+ 2)kg(x, x). Since (M, g) is assumed to be an Einstein manifold the previous equation becomes ©b(n+ 4) + 3cªηxx =−n2(n+ 2)akRk2+bkRk2+2cτ2 n2−2(n+ 2)kogxx and taking traces this gives ©b(n+ 4) + 3cªkRk2=−nn2(n+ 2)akRk2+bkRk2+2cτ2 n2−2(n+ 2)ko. The last two equations and the fact that b(n+ 4) + 3c6= 0 imply ηxx = ( kRk2/n)gxx and thus polarization gives η= ( kRk2/n)g. Hence, it follows from the assumption that ωxxxx =−1 cµna +b nkRk2−k¶g2 xx which shows that (M, g) is 2–stein. Proposition 4.11. Let (Mn, g)a Riemannian manifold and Wa simple Weyl invariant of degree 2ν,ν > 1, such that AW(n−1) 6= 0, (2n+ 1)νAW(n−1) −20(n+ 12)B1 W(n−1) + 5n2B2 W(n−1) 6= 0, (2n+ 1)νAW(n−1) + 40nB1 W(n−1) + 5n2B2 W(n−1) 6= 0, If the corresponding Weyl invariants of geodesic spheres f W(expm(ru)) depend neither on the center mnor on the direction u, then Mis 2–stein. Proof. Since AW(n−1) 6= 0, using the coefficient f W2(u) given in Theorem 4.7, we get that τ−2ρuu(n+1)/3 is independent of mand u. This implies that the manifold Mis Einstein. Now, the coefficient f W4is also constant by hypothesis. Using the fact that Mis Einstein, we obtain B1 W(n−1) kRk2−4B1 W(n−1) n X a,b,c=1 R2 uabc −µ2n+ 1 45 νAW(n−1) −4(n+ 12) 9B1 W(n−1) −n2 9B2 W(n−1)¶n X a,b=1 R2 uaub =constant. 4.2.2 Total scalar curvatures of geodesic spheres 77 The above equation has the form of that of Lemma 4.10. The last two conditions of Proposition 4.11 ensure that the latter lemma can be applied and the result follows. Remark 4.12.Let Wbe a simple Weyl invariant such that AW(n−1) 6= 0. If Mis a Riemannian manifold such that the corresponding Weyl invariants of geodesic spheres f W(expm(ru)) depend only on the radius, then the above proof shows that Mis an Einstein manifold. Remark 4.13.It was proved in [33] that if eτ(expm(ru)) depends neither on the center m nor on the direction u, then the manifold is 2–stein. Moreover, if the manifold is assumed to be analytic, then it is harmonic. Example 4.14.It can easily be shown that the conditions in Proposition 4.11 hold for all the curvature invariants of Example 4.9. Hence, those may be used to characterize 2–stein manifolds. Proposition 4.15. Let (M, g)be a complete analytic Riemannian manifold with constant Weyl invariants and such that all its small geodesic spheres have constant scalar curvature. Then, Mis locally isometric to a two–point homogeneous manifold or a Damek–Ricci space. Proof. As all the Weyl invariants of Mare constant, Mis locally homogeneous [113]. Since all the small geodesic spheres have constant scalar curvature and the manifold is analytic, Mis harmonic [33]. Complete homogenous harmonic manifolds have been classified in [87]. According to this paper, Mis locally isometric to a two–point homogeneous manifold or a Damek–Ricci space. Corollary 4.16. Let (M, g)be a complete analytic Riemannian manifold with constant Weyl invariants such that all its small geodesic spheres have also constant Weyl invariants. Then, Mis locally isometric to a two–point homogeneous space. Proof. Using the previous proposition, Mis locally isometric to a two–point homogeneous space or a Damek–Ricci space. On the other hand, all the small geodesic spheres of M are homogeneous, as they also have constant Weyl invariants. Hence, Mis Osserman [80]. But a Damek–Ricci space cannot be Osserman unless it is symmetric [17]. The result follows because locally symmetric Damek–Ricci spaces are locally isometric to a two–point homogeneous space. 4.2.2 Total scalar curvatures of geodesic spheres Since a geodesic sphere is a compact Riemannian submanifold, one may consider the integral of a curvature invariant Wfor geodesic spheres. Following, for example, [33], we define the total scalar curvature Wassociated with the scalar curvature invariant Wby W(m, r) = ZGm(r)f W=rn−1ZSn−1 (f Wθm)(expm(ru))du, 78 4 The Riemannian setting where f Wis the corresponding curvature invariant of Gm(r), θmis the volume density function at mand du is the volume element of Sn−1. Then, Wis a function depending on the base point and the radius of the geodesic sphere. Example 4.17.When (M, g) is a Riemannian manifold of constant sectional curvature λ > 0, each geodesic sphere Gm(r) has constant sectional curvature e λ=λ/ sin2r√λ[83] (here, we only consider the positive curvature case; similar expressions can be obtained for negative and zero curvature). We now compute the total scalar curvature associated with a Weyl invariant Wof degree 2ν. From Remark 4.2 we get f W= (n−1)(n−2)AW(n−1) µλ sin2r√λ¶ν . In a space of constant sectional curvature λ > 0 the volume density function is θm(expm(ru)) = Ãsin r√λ r√λ!n−1 (see for example [82], [128]) and we have the exact expression for the total scalar curvature associated with W ZGm(r)f W=cn−1(n−1)(n−2)AW(n−1) Ãsin r√λ √λ!n−1−2ν . We emphasize that the above total scalar curvature does not depend on the base point m. In order to obtain a power series expansion of a total scalar curvature we need the volume density function of a Riemannian manifold [33]. Lemma 4.18. Let θmbe the volume density function at a point m. Then we have θm(expm(ru)) = s X α=0 rαθα(u) + O¡rs+1¢, where θα(u),α≥2, is a partial curvature invariant of degree αin the direction of uwith αdegrees of freedom. The first terms of this power series expansion are θm(expm(ru)) = 1 −1 6ρuu(m)r2−1 12∇uρuu(m)r3 +³−1 180 n X a,b=1 R2 uaub +ρ2 uu 72 −1 40∇2 uuρuu´(m)r4+O¡r5¢. Proof. First, we consider the mean curvature function of a geodesic sphere which is easily obtained from Lemma 4.4 taking traces hm(expm(ru)) = s−1 X α=−1 rαhα+1(u) + O(rs), 4.2.2 Total scalar curvatures of geodesic spheres 79 where hα(u), α≥0 are partial scalar curvature invariants of degree αin the direction of uand with αdegrees of freedom. The first terms in the power series expansion of hmare well–known [33], [128] hm(expm(ru)) =(n−1 r−r 3ρuu−r2 4∇uρuu −r3Ã1 45 n X a,b=1 R2 uaub +1 10∇2 uuρuu !)(m)+O¡r4¢. Now, using the relation hm(expm(ru)) = n−1 r+∂ ∂r log θm(expm(ru)) , we obtain θα(u) = [α/2] X β=1 1 β! X γ1+···+γβ=α hγ1(u)···hγβ(u) γ1···γβ , α ≥2, where [ ] denotes the integer part of a real number. The result follows from the above formula after considering the properties and explicit expressions of the terms hα. Using standard arguments of calculus one can show that the volume of a Euclidean sphere of radius 1 in Rnis given by cn−1=n πn 2 Γ¡n 2+ 1¢, where Γ is the gamma function defined by Γ(α) = R∞ 0e−ttα−1dt =R∞ −∞ e−t2|t|2α−1dt. The following result is a technical lemma which will be used in the proof of the Theorem 4.20. We only point out the main steps of the proof. Lemma 4.19. Let ωbe a covariant tensor of order 2ν. Then, ZSn−1 ωu···udu =cn−1 2νν! ν−1 Y α=0 (n+ 2α) n X α1···α2ν=1 δα1α2···δα2ν−1α2νX σ∈G2ν ωασ(1)···ασ(2ν). where G2νis the group of permutations of 2νelements. Proof. We proceed by induction. If ν= 1, the expression of Lemma 4.19 is a well–known fact. See, for example, [83]. Next, let ωbe a covariant tensor of order 2(ν+ 1). Choose an orthonormal basis {ei}at the origin of Rnand write the unit vector uwith respect to that basis as u=Pixiei. Then ZSn−1 ωu···udu = n X α1···α2ν+2=1 ωα1···α2ν+2 ZSn−1 xα1···xα2ν+2 du. 80 4 The Riemannian setting We recall the formula for integrating polynomials along Euclidean spheres [82]: ZSn−1 xβ1 1···xβn ndu =cn−1 β1)···βn) n(n+ 2) ···(n+β1+···+βn+ 2), where 2β) = (2β−1)(2β−3) ···3·1,2β−1) = 0,for all β∈Nand 0) = 1. Concentrating on the last index α2ν+2 and using the previous formula, we get ZSn−1 ωu···udu = n X α1...α2ν+1=1 2ν+1 X β=1  1 P2ν+1 γ=1 δαβαγ ωα1...α2ν+1 ↓ β ...α2ν+1 ZSn−1 xα1. . . xα2νx2 α2ν+1 du  =1 n+ 2ν n X α=1 2ν+1 X β=1  ZSn−1 ω(u, . . . , u, eα ↓ β , u, . . . , u, eα)du . Now the inner integrand is a tensor of order 2νand we can apply our induction hypothesis to get ZSn−1 ωu···udu=cn−1 2νν! ν Y α=0 (n+ 2α) n X α1...α2ν+1=1 2ν+1 X β=1µδα1α2···δα2ν−1α2νX σ∈G2ν ωασ(1)...α2ν+1 ↓ β ...ασ(2ν)α2ν+1 ¶ =cn−1 2νν! ν Y α=0 (n+ 2α) n X α1...α2ν+2=1Ãδα1α2···δα2ν+1α2ν+2 X σ∈G2ν+2 σ(2ν+2)=2ν+2 ωασ(1)...ασ(2ν+2) !, from where the result follows. Theorem 4.20. Let Wbe a simple Weyl invariant of degree 2ν. The total scalar curvature associated with Whas a power series expansion W(m, r) = ZGm(r)f W=cn−1rn−1−2ν  [s/2] X α=0 r2αW2α(m) Qα−1 β=0(n+ 2β)+O¡rs+1¢ , where W2α(m),α≥1, is a scalar curvature invariant of Mat mof degree α,[ ] denotes the integer part of a real number and W0(m)=(n−1)(n−2)AW(n−1), W2(m) = −(n−2)(n−2ν−1)AW(n−1) 6τ(m), W4(m) = µC1 W(n−1) kRk2+C2 W(n−1) kρk2+C3 W(n−1) τ2 −(n−2)(n−2ν−1)AW(n−1) 20 ∆τ¶(m). 4.2.2 Total scalar curvatures of geodesic spheres 81 Moreover, we have the relation 2C1 W(n−1) + (n−1)C2 W(n−1) + n(n−1)C3 W(n−1) =(n−2)(n+ 2)(n−1−2ν)(5n−10ν−7) 360 AW(n−1). Proof. By definition we have W(m, r) = ZGm(r)f W=rn−1ZSn−1³f Wθm´(expm(ru))du. Using Theorem 4.7 and Lemma 4.18, we get ³f Wθm´(expm(ru)) = s−2ν X α=−2ν rα¯ Wα+2ν(u) + O¡rs−2ν+1¢, where ¯ Wα(u), α≥2, is a partial curvature invariant in the direction of uwith degree α such that for all the Weyl invariants involved in its construction their number of degrees of freedom has the same parity as α. In fact, we have ¯ Wα(u) = Pα β=0 f Wβ(u)θα−β(u), α≥2, and in particular, using Theorem 4.7 and Lemma 4.18 ¯ W0(u) = (n−1)(n−2)AW(n−1), ¯ W1(u) = 0, ¯ W2(u) = AW(n−1) µντ −4ν(n+ 1) + (n−1)(n−2) 6ρuu¶(m), ¯ W4(u) = ¯ω4(u) + AW(n−1) 2µν∇2 uuτ−40ν(n+3)+(n−1)(n−2) 20 ∇2 uuρuu¶(m), where ¯ω4(u) is a simple directional curvature invariant of degree 4. Integration gives W(m, r) = rn−1Ãs−2ν X α=−2ν rαZSn−1 ¯ Wα+2ν(u)du +O¡rs−2ν+1¢!. If αis odd, ¯ Wα+2ν(u) is a linear combination of Weyl invariants in the direction of uwith an odd number of degrees of freedom. Each one of them is an odd function on a sphere, and thus its integral vanishes. Hence, we have W(m, r) = rn−1   [s−2ν 2] X α=−ν r2αZSn−1 ¯ W2α+2ν(u)du +O¡rs−2ν+1¢  . The problem of integrating ¯ W2α(u), with α≥1, reduces to the integration of directional Weyl invariants of degree 2αin the direction of uwith an even number of degrees of freedom. 88 4 The Riemannian setting (iv) For each sufficiently small geodesic sphere, RGm(r) ˇ e Ris the same as in the two–point homogeneous space (n6= 7). (v) Mis locally isometric to that two–point homogeneous space. Remark 4.31.If the manifold is assumed to be Einstein or locally conformally flat then all the simple Weyl invariants of degree 4 and 6 appearing in (4.1) and (4.2) can be used to characterize the two–point homogeneous spaces. Dimension 5 (resp. 7) is a singular case for simple scalar curvature invariants of degree 4 (resp. 6) since the corresponding total curvatures of geodesic spheres are topological invariants in manifolds of constant sectional curvature (see Remark 4.27). Nonetheless, we can still detect constant curvature manifolds [37], [44] although we cannot determine the exact value of their sectional curvature. Theorem 4.32. Let Ma5–dimensional Riemannian manifold. The following statements are equivalent: (i) For each sufficiently small geodesic sphere, RGm(r)ke Rk2is the same as in a manifold of constant sectional curvature. (ii) For each sufficiently small geodesic sphere, RGm(r)keρk2is the same as in a manifold of constant sectional curvature. (iii) Mhas constant sectional curvature. Proof. Using Theorem 4.20 and the formulas for C1 kRk2and C2 kRk2at the beginning of this section we get ZGm(r)ke Rk2= 24c4+r4 35c4½50 3µkRk2−1 2kρk2¶+35 9µkρk2−1 5τ2¶¾(m) + O¡r6¢. By Remark 4.27 we obtain that in a manifold of constant sectional curvature RGm(r)ke Rk2= 24c4. Hence, comparing this two expressions we get µkRk2−1 2kρk2¶+35 9µkρk2−1 5τ2¶= 0, and it follows from Lemma 4.1 that Mhas constant sectional curvature. For kρk2we proceed in an analogous way taking into account that ZGm(r)keρk2= 36c4+r4 35c4½5 3µkRk2−1 2kρk2¶+298 18 µkρk2−1 5τ2¶¾(m) + O¡r6¢. and using Lemma 4.1 once again. In a similar way we obtain the following result. We delete the details [38]. 4.3.2 Total scalar curvatures of boundaries of geodesic disks 89 Theorem 4.33. Let Ma7–dimensional Riemannian manifold. The following statements are equivalent: (i) For each sufficiently small geodesic sphere, RGm(r)eτke Rk2is the same as in a manifold of constant sectional curvature. (ii) For each sufficiently small geodesic sphere, RGm(r)ˇ eρis the same as in a manifold of constant sectional curvature. (iii) For each sufficiently small geodesic sphere, RGm(r)heρ⊗eρ, ¯ e Riis the same as in a manifold of constant sectional curvature. (iv) For each sufficiently small geodesic sphere, RGm(r)heρ, ˙ e Riis the same as in a manifold of constant sectional curvature. (v) For each sufficiently small geodesic sphere, RGm(r) ˇ e Ris the same as in a manifold of constant sectional curvature. (vi) Mhas constant sectional curvature. 4.3.2 Total scalar curvatures of boundaries of geodesic disks Geodesic disks were introduced by O. Kowalski and L. Vanhecke as a generalization of two–dimensional disks in the Euclidean space R3. In a series of papers [93], [94], [95], they investigated their volume properties in relation to local homogeneity and gave a characterization of two–point homogeneous spaces by means of the volumes of their small geodesic disks. Since the boundaries of small geodesic disks are compact submanifolds, we are interested in their total scalar curvatures obtained by integrating the corresponding scalar curvature invariants. The geodesic disk,¯ Dξ m(r), of sufficiently small radius r, centered at m∈Mand orthogonal to ξ∈TmM, is defined by ¯ Dξ m(r) = {expm(su) : u∈TmM, kuk= 1, g(u, ξ) = 0,0≤s≤r} ={p∈M:d(m, p)≤r}∩expm(Rξ⊥). For the purpose of this section and the investigation of total scalar curvatures, we consider the boundaries of geodesic disks Dξ m(r) = {p∈M:d(m, p) = r}∩expm(Rξ⊥). The boundary of a geodesic disk is nothing but a geodesic sphere of the (local) manifold expm(Rξ⊥) centered at mfor sufficiently small radius. Throughout this section we use the following notation. The objects of Mare denoted by R,ρand so on, the objects of expm(Rξ⊥) are denoted by e R,eρ,. . . and the objects of boundaries of geodesic disks are denoted by ˆ R, ˆρand so on. 90 4 The Riemannian setting In what follows we consider the total scalar curvatures of boundaries of geodesic disks associated with scalar curvature invariants of degree 2 and 4. Hence, it suffices to study the scalar curvature τand the scalar curvature invariants in (4.1). We do not consider the Laplacian of the scalar curvature since RGm(r)e ∆eτdu = 0 by the divergence theorem. In order to obtain the first terms in the power series expansions of total curvatures of boundaries of geodesic disks, we need the following result relating scalar curvature invariants of degree 2 and 4 of expm(Rξ⊥) with the corresponding objects in M. Lemma 4.34. Let (M, g)be an n–dimensional Riemannian manifold and ξ∈TmMa unit vector. Then, the following relation holds at m: ke Rk2=kRk2+ 4 n X i,j=1 R2 ξiξj −4 n X i,j,k=1 R2 ξijk , keρk2=kρk2+ρ2 ξξ −2 n X i=1 ρ2 ξi + n X i,j=1 R2 ξiξj −2 n X i,j=1 ρijRξiξj , eτ=τ−2ρξξ , e ∆eτ= ∆τ−2∆ρξξ + 2∇2 ξξρξξ −∇2 ξξτ+4 9ρ2 ξξ −4 9 n X i=1 ρ2 ξi +4 3 n X i,j=1 R2 ξiξj −2 3 n X i,j,k=1 R2 ξijk. Proof. It follows from the work in [93], after some calculations. The first terms in the power series expansions of the total curvatures of the boundaries of geodesic disks are obtained from the corresponding ones of geodesic spheres. We use the results of Section 4.3.1 in conjunction to Lemma 4.34. We omit the calculations which are straightforward and immediately state the different expansions separately in the following Proposition 4.35. Let (M, g)be an n-dimensional Riemannian manifold, m∈Mand ξ∈TmMa unit vector. Then, for sufficiently small radius r, one has the following power series expansions ZDξ m(r) ˆτ=cn−2rn−2((n−2)(n−3) r2−(n−3)(n−4) 6(n−1) (τ−2ρξξ) +r2 (n−1)(n+ 1)Ãn2−9n+ 2 72 τ2−(n+ 2)(n+ 1) 120 kRk2+n2+ 3n+ 17 45 kρk2 −(n−3)(n−4) 20 ∆τ+(n−3)(n−4) 20 ¡∇2 ξξτ−2∇2 ξξρξξ + 2∆ρξξ¢−n2−9n+ 2 18 τρξξ −(n+ 2)(n+ 11) 45 n X i=1 ρ2 ξi −(n−4)(7n−11) 90 n X i,j=1 R2 ξiξj +n2−2n+ 7 15 n X i,j,k=1 R2 ξijk −2(n2+ 3n+ 17) 45 n X i,j=1 Rξiξjρij +(n−1)(n−4) 18 ρ2 ξξ!+O¡r3¢)(m), 4.3.2 Total scalar curvatures of boundaries of geodesic disks 91 ZDξ m(r) ˆτ2=cn−2rn−2½(n−2)2(n−3)2 r4−(n−3)2(n−6)(n−2) 6(n−1)r2(τ−2ρξξ) +1 (n−1)(n+ 1)µn4−18n3+ 77n2−164n−84 72 τ2−(n−3)2(n−6)(n−2) 20 ∆τ +n4+ 6n3+ 19n2−74n+ 168 45 kρk2−(n−2)(n−3)(n2+ 11n−2) 120 kRk2 −n4+ 26n3−31n2−4n+ 228 45 n X i=1 ρ2 ξi +(n−2)(n−3)(n2+n+ 8) 15 n X i,j,k=1 R2 ξijk −n4−18n3+ 77n2−164n−84 18 τρξξ −7n4−78n3+ 223n2−488n+ 276 90 n X i,j=1 R2 ξiξj −2(n4+ 6n3+ 19n2−74n+ 168) 45 n X i,j=1 Rξiξjρij +n4−10n3+ 57n2−136n−60 18 ρ2 ξξ +(n−3)2(n−6)(n−2) 20 ¡∇2 ξξτ−2∇2 ξξρξξ + 2∆ρξξ¢¶+O(r)¾(m), ZDξ m(r)kˆρk2=cn−2rn−2½(n−2)(n−3)2 r4−(n−3)2(n−6) 6(n−1)r2(τ−2ρξξ) +1 (n−1)(n+ 1)µn3−16n2+ 13n+ 46 72 τ2−n3−12n2+ 5n−14 120 kRk2 +n3+ 28n2−75n−74 45 kρk2−(n−3)2(n−6) 20 ∆τ+n3−35n+ 38 18 ρ2 ξξ +(n−3)2(n−6) 20 ¡∇2 ξξτ−2∇2 ξξρξξ + 2∆ρξξ¢−n3−16n2+ 13n+ 46 18 τρξξ −2(n3+ 28n2−75n−74) 45 n X i,j=1 Rξiξjρij −7n3−164n2+ 435n−218 90 n X i,j=1 R2 ξiξj −n3+ 68n2−195n−94 45 n X i=1 ρ2 ξi +n3−12n2+ 25n−34 15 n X i,j,k=1 R2 ξijk¶+O(r)¾(m), ZDξ m(r)kˆ Rk2=cn−2rn−2½2(n−2)(n−3) r4−(n−3)(n−6) 3(n−1)r2(τ−2ρξξ) +1 (n−1)(n+ 1)µn2−13n+ 14 36 τ2+59n2−211n+ 142 60 kRk2−(n−3)(n−6) 10 ∆τ +2(n2−39n+ 98) 45 kρk2+173n2−657n+ 514 45 n X i,j=1 R2 ξiξj −4(n2−39n+ 98) 45 n X i,j=1 Rξiξjρij −2(29n2−101n+ 62) 15 n X i,j,k=1 R2 ξijk +(n−3)(n−6) 10 ¡∇2 ξξτ−2∇2 ξξρξξ + 2∆ρξξ¢ −2(n2−69n+ 178) 45 n X i=1 ρ2 ξi −n2−13n+ 14 9τρξξ +(n−2)(n−23) 9ρ2 ξξ¶+O(r)¾(m). 92 4 The Riemannian setting As an application of the expansions in Theorem 4.35 we are now ready to obtain characterizations of the two–point homogeneous spaces by means of the total curvatures of the boundaries of geodesic disks. Lemma 4.36. Let Mbe an n–dimensional Riemannian manifold. Assume that one of the following holds: (i) We have n > 4and RDξ m(r)ˆτcoincides with the corresponding one in an Einstein manifold. (ii) We have 3< n 6= 6 and any of RDξ m(r)ˆτ2,RDξ m(r)kˆρk2or RDξ m(r)kˆ Rk2coincides with the corresponding one in an Einstein manifold. Then Mis an Einstein manifold with the same scalar curvature as the model space. Proof. In case (i) it follows from the constant term of the power series expansion of RDξ m(r)ˆτ in Proposition 4.35. In case (ii) it follows from the coefficient of r−2in the power series expansion of RDξ m(r)ˆτ2,RDξ m(r)kˆρk2or RDξ m(r)kˆ Rk2in Proposition 4.35. Lemma 4.37. Let Mbe an n–dimensional Riemannian manifold. Assume that one of the following holds: (i) We have n > 4and RDξ m(r)ˆτdoes not depend on the normal direction ξ. (ii) We have 3< n 6= 6 and one of RDξ m(r)ˆτ2,RDξ m(r)kˆρk2or RDξ m(r)kˆ Rk2does not depend on the normal direction ξ. Then, Mis 2–stein. Proof. Assume that (i) holds. Since the total scalar curvatures of the boundaries of geodesic disks of the manifold Mdo not depend on the normal direction, the constant term and the coefficient of r2in its power series expansion given by Proposition 4.35 are independent of the unit ξ∈TM. From the independent term it follows that τ−2ρξξ is constant and hence Mis an Einstein space. Moreover, for an Einstein manifold, the coefficient of r2in the power series expansion of RDξ m(r)ˆτbecomes 1 (n−1)(n+ 1)½(n−4)(5n3−37n2+ 62n+ 92) 360n2τ2−(n+ 2)(n+ 1) 120 kRk2 −(n−4)(7n−11) 90 n X i,j=1 R2 ξiξj +n2−2n+ 7 15 n X i,j,k=1 R2 ξijk¾. Therefore, using Lemma 4.10 we get that Mis a 2–stein space. 4.3.2 Total scalar curvatures of boundaries of geodesic disks 93 If (ii) holds, we get the result in a similar way. From the coefficient of r−2in the power series expansions of RDξ m(r)ˆτ2,RDξ m(r)kˆρk2and RDξ m(r)kˆ Rk2we deduce that Mis Einstein, and then, rewriting the independent terms of those power series expansions we get 1 (n−1)(n+ 1)½5n6−102n5+ 789n4−2712n3+ 3352n2+ 1520n−5712 360n2τ2 −(n−2)(n−3)(n2+ 11n−2) 120 kRk2−7n4−78n3+ 223n2−488n+ 276 90 n X i,j=1 R2 ξiξj +(n−2)(n−3)(n2+n+ 8) 15 n X i,j,k=1 R2 ξijk¾, 1 (n−1)(n+ 1)½−7n3−164n2+ 435n−218 90 n X i,j=1 R2 ξiξj −n3−12n2+ 5n−14 120 kRk2 +5n5−92n4+ 605n3−1622n2+ 548n+ 2696 360n2τ2+n3−12n2+ 25n−34 15 n X i,j,k=1 R2 ξijk¾, 1 (n−1)(n+ 1)½5n4−77n3+ 14n2+ 1180n−2072 180n2τ2+59n2−211n+ 142 60 kRk2 +173n2−657n+ 514 45 n X i,j=1 R2 ξiξj −2(29n2−101n+ 62) 15 n X i,j,k=1 R2 ξijk¾, for RDξ m(r)ˆτ2,RDξ m(r)kˆρk2and RDξ m(r)kˆ Rk2, respectively. Applying Lemma 4.10 one eventually gets the result. Now we are ready to derive the desired characterizations of the two–point homogeneous spaces for n > 4. Theorem 4.38. Let Mbe an n–dimensional Riemannian manifold whose holonomy group is contained in the holonomy group of a two–point homogeneous space. If n > 4and RDξ m(r)ˆτ coincides with that of the two–point homogeneous space for sufficiently small radius, then Mis locally isometric to that two–point homogeneous space. Proof. It follows from Lemma 4.37 (i) that Mis 2–stein and thus super–Einstein [33], from where we get that n X i,j=1 R2 ξiξj =1 n(n+ 2) µ3 2kRk2+1 nτ2¶and n X i,j,k=1 R2 ξijk =1 nkRk2. Then, the coefficient of r2in the power series expansion of RDξ m(r)ˆτgiven by Proposition 4.35 becomes n−4 n(n2−1)(n+ 2) ½5n4−27n3−12n2+ 188n+ 228 360nτ2−n3+n2+ 26n+ 6 120 kRk2¾. 94 4 The Riemannian setting Now the result is obtained just by comparing this with the corresponding coefficient in the model space and using Lemma 4.1. It is worthwhile to emphasize that dimension four is excluded in the previous theorem. Since the boundaries of the geodesic disks in a 4–dimensional manifold are compact surfaces, the total curvature RDξ m(r)ˆτis the Gauss Bonnet integral, and thus a topological invariant. Theorem 4.39. Let Mbe an n–dimensional Riemannian manifold whose holonomy group is contained in the holonomy group of a two–point homogeneous space. If 3< n 6= 6 and one of RDξ m(r)ˆτ2,RDξ m(r)kˆρk2or RDξ m(r)kˆ Rk2coincides with that of the two–point homogeneous space for sufficiently small radius, then Mis locally isometric to that two–point homogeneous space. Proof. We proceed as in the previous theorem. Using Lemma 4.37 (ii) we get that Mis 2– stein. Then, the independent terms of the power series expansions of RDξ m(r)ˆτ2,RDξ m(r)kˆρk2 or RDξ m(r)kˆ Rk2given by Proposition 4.35 become 1 n(n2−1)(n+ 2)½5n7−92n6+ 585n5−1162n4−1760n3+ 7332n2−720n−12528 360nτ2 −n6−9n4−190n3+ 714n2−840n−216 120 kRk2¾, n−3 n(n2−1)(n+ 2)½5n5−67n4+ 220n3+ 220n2−1380n−2088 360nτ2 −(n2−14n−2)(n2−n+ 18) 120 kRk2¾, 1 n(n2−1)(n+ 2)½(n−3)(5n4−52n3−296n2+ 1012n+ 696 180nτ2 +(n−3)(59n3−148n2−34n−12) 60 ¾, respectively. Now the result follows by comparing these coefficients with the corresponding ones in the model spaces and using Lemma 4.1. Chapter 5 Geodesic celestial spheres in Lorentzian manifolds In the previous chapter we have seen that every Riemannian manifold carries a so–called Riemannian distance function whose level sets with respect to a point are exactly the geodesic spheres of the manifold. Moreover, geodesic spheres can also be seen as the image by the exponential map of Euclidean spheres in the tangent space at a point. In the general semi–Riemannian setting, such a distance is not defined and the fact that the pseudo–spheres of the tangent space are not compact leads us to believe that their image by the exponential map is not suitable for study. A distance–like function d:M×M→[0,∞] may be defined for space–times. For any p, q ∈Mwe have d(p, q) = 0 if and only if qis not in the causal future of pand d(p, q) = sup{L(c) : cis a future directed non–spacelike curve from pto q}if qis in the causal future of p. However, the properties of this distance function are completely different from those in the Riemannian setting [7]. For example, the “Lorentzian distance” may fail to be continuous or finite–valued and its level sets with respect to a given point are not compact. Some properties of those sets have been previously investigated [4], [56], but they do not seem to be adequate for the investigation of volume properties. Therefore, different families of objects have been considered for this purpose in Lorentzian geometry. We give an overview of these constructions in Section 5.1. The concept of geodesic celestial sphere which we introduce in this chapter is somehow an extension of the concept of geodesic disk to Lorentzian geometry. Given a unit timelike vector, its orthogonal complement in the tangent space has definite signature. In Relativity, a unit timelike vector ξis called an instantaneous observer and its orthogonal complement in the tangent space is called the infinitesimal rest–space of ξ. In Special Relativity the rest space of an instantaneous observer corresponds to the Newtonian universe perceived by this observer. A geodesic celestial sphere is the image by the exponential map of a sphere centered at the origin of the rest–space associated with certain instantaneous observer. It is clear that every geodesic celestial sphere (for sufficiently small radius) is a compact Riemannian submanifold. We see in this chapter that geodesic celestial spheres are closely related to geodesic spheres and they inherit most of their properties. Thus, volume 95 96 5 Geodesic celestial spheres in Lorentzian manifolds properties can be discussed and in many cases we get analogous results to those of the Riemannian setting. This chapter is organized as follows. Section 5.1 reviews some constructions in Riemannian and Lorentzian geometry which we attempt to generalize. In Section 5.2 we study the volume of geodesic celestial spheres and give the necessary background to prove the main theorems of this section, namely, Theorems 5.16 and 5.17 (which compare the volumes of sufficiently small geodesic celestial spheres in a Lorentzian manifold with the corresponding ones in a Lorentzian space form) and Theorem 5.22 (which characterizes locally isotropic space–times). Finally, in Section 5.3 we state some results analogous to Theorem 5.22, showing that local isotropy can be detected by considering the total curvatures of geodesic celestial spheres. We also give examples of scalar curvature invariants which may be used for this characterization. 5.1 Volume comparison results Any Riemannian manifold (Mn+1, g) carries a Riemannian distance function which has a very nice behavior with respect to the underlying structure of the manifold. Therefore, a natural family of subregions of a Riemannian manifold to be considered is that defined by the level sets of the Riemannian distance function with respect to a base point (that is, geodesic spheres) or with respect to some topologically embedded submanifolds (that is, tubes around a submanifold). For sufficiently small radii r > 0, geodesic spheres Gm(r) are obtained by projecting the Euclidean spheres Sn(r) centered at 0 ∈TmMvia the exponential map. Therefore, they are a nice family of hypersurfaces and their volume can be calculated as S(m, r) = vol (Gm(r)) = rnZSn θm(expm(ru))du. Comparison theorems for the volumes of subregions of Riemannian manifolds under some curvature hypotheses have played an important role in Riemannian geometry. For instance, the Bishop–G¨unther inequalities show lower (resp. upper) bounds for volumes of geodesic balls and tubes by imposing upper (resp. lower) bounds on the sectional curvature. These inequalities have been improved by assuming weaker conditions on the Ricci tensor or by considering the ratio between the volumes of geodesic balls in the manifold and the model spaces (see for example [82] and the references therein). The basic idea behind the Bishop–G¨unter and Gromov comparison theorems [82], is that under suitable curvature conditions the Riccati differential equation S0+S2+Ru= 0. becomes an inequality and its solutions give upper or lower bounds for the volume density function θmin terms of the corresponding function in the model space via hm(expm(ru)) = n r+∂ ∂r log θm(expm(ru)). 5.1 Volume comparison results 97 Finally, an integration process from the Riccati equations leads to [18], [86] Theorem 5.1. Let (Mn+1, g)be a complete Riemannian manifold and assume that ris not greater than the distance between mand its cut locus. Let KMdenote the sectional curvature of (M, g). (i) If KM≥λ, then volM(Gm(r)) ≤volM(λ)(G˜m(r)). (ii) If KM≤λ, then volM(Gm(r)) ≥volM(λ)(G˜m(r)). Here, M(λ)is a model space of constant sectional curvature λand ˜m∈M(λ). Moreover, equalities hold for (i) or (ii) and some radii if and only if Gm(r)is isometric to the corresponding geodesic sphere in the model space. A sharper result involving the Ricci curvature instead of the sectional curvature was proved by R. L. Bishop [18]. Theorem 5.2. Let (Mn+1, g)be a complete Riemannian manifold. Assume that ris not greater that the distance between mand its cut locus and the Ricci curvature ρMof (M, g) satisfies ρM(v, v)≥n λ for all vectors v∈TM. Then volM(Gm(r)) ≤volM(λ)(G˜m(r)), where M(λ)is a space of constant sectional curvature λ. The equality holds if and only if Gm(r)is isometric to the corresponding geodesic sphere in the model space. A further generalization of Theorem 5.2 was obtained by M. Gromov as follows [85]. Theorem 5.3. Let (Mn+1, g)be a complete Riemannian manifold. Assume that ris not greater that the distance between mand its cut locus and that the Ricci curvature ρMof (M, g)satisfies ρM(v, v)≥n λ for all vectors v. Then the function r7→ volM(Gm(r)) volM(λ)(G˜m(r)), where M(λ)is a space of constant sectional curvature λ, is non–increasing. When the attention is turned from Riemannian manifolds to space–times, various difficulties emerge. For example, conditions on bounds for the sectional curvature (resp. the Ricci tensor) easily produce manifolds of constant sectional curvature (resp. Einstein) [7], [108]. This demands a revision of such conditions [5] (see Section 5.2.2). However, a more difficult task is related to the consideration of the regions under investigation. This is mainly due to the fact that when dealing with general semi–Riemannian manifolds there is no “semi–Riemannian distance function”. In fact, a distance–like function is only defined for space–times, but even in this case its properties are completely different from those in the Riemannian setting (see [7]). For instance, level sets of the Lorentzian distance function with respect to a given point are not compact and they do not seem to be adequate for the investigation of volume properties. Therefore, different families of objects have been 104 5 Geodesic celestial spheres in Lorentzian manifolds the instantaneous observer ξ∈TmN(λ). Thus, in this case we can use the unambiguous notation voln−1(G(r)) = voln−1¡Gξ m(r)¢. For the purpose of the comparison results below, we will also denote by volM n−1¡Gξ m(r)¢the (n−1)–dimensional volume of the geodesic celestial sphere Gξ m(r) of radius rand center massociated with the instantaneous observer ξin the manifold M. In the following subsection we give the technical background to prove the main theorems of this chapter, which are in Subsections 5.2.2 and 5.2.3. 5.2.1 Power series expansions The technique we use to prove the main results of this chapter relies on the possibility of writing down the first terms in the power series expansion of the function r7→ volM n−1¡Gξ m(r)¢, for sufficiently small r. From now on we assume the following notation. For fixed m∈Mand ξ∈TmMwe consider the Riemannian submanifold f M= expm(U∩ξ⊥), where Uis a sufficiently small neighborhood of m. Objects of f Mare denoted by e. We choose an orthonormal basis {e0=ξ, e1, . . . , en}at m. Lemma 5.10. With the above notation, the first and second order curvature invariants of Mand f Mat the base point msatisfy ke Rk2=kRk2+ 4 n X i,j,k=1 R2 ξijk −4 n X i,j=1 R2 ξiξj, keρk2=kρk2+ 2 n X i=1 ρ2 ξi −ρ2 ξξ + n X i,j=1 R2 ξiξj + 2 n X i,j=1 ρijRξiξj, eτ=τ+ 2ρξξ, e ∆eτ= ∆τ+ 2∆ρξξ +∇2 ξξτ+ 2∇2 ξξρξξ +4 9 n X i=1 ρ2 ξi +2 3 n X i,j,k=1 R2 ξijk. Proof. Denote by ξa local extension of ξ∈TmMto the normal bundle of f M. If cis a radial geodesic in f Mstarting at m, then e ∇c0c0=∇c0c0=II(c0, c0) = 0 since f M= expm(U∩Rξ⊥). Thus, taking covariant derivatives and evaluating at m, we get e ∇k u···uσuu = 0, k≥0, for all u∈Tmf M. For k= 0 we immediately get by polarization that σuv = 0,for all u, v ∈Tmf M. Now put k= 1 and take arbitrary a, b, c ∈Rand u, v, w ∈Tmf M. We have 0 = e ∇au+bv+cwσau+bv+cw,au+bv+cw =···+ 2abc ³e ∇uσvw +e ∇vσuw +e ∇wσuv´+··· 5.2.1 Power series expansions 105 and hence e ∇uσvw +e ∇vσuw +e ∇wσuv = 0. Then it follows from the Riccati equation that Ruvwξ =e ∇uσvw −e ∇vσuw and Ruwvξ =e ∇uσvw −e ∇wσuv. Therefore, we can express e ∇σin terms of the curvature tensor of the ambient manifold M as follows e ∇uσvw =1 3(Ruvwξ +Ruwvξ),for all u, v, w ∈Tmf M. We now determine the curvature tensor of f Mat m. An immediate application of the Gauss equation and σuv = 0 shows that (5.1) e Rxyvw =Rxyvw, for all x, y, v, w ∈Tmf M. Taking covariant derivatives in the Gauss equation we get (5.2) e ∇Ze RXY V W =∇ZRXY V W +σZX RξY V W +σZY RXξV W +σZV RXY ξW +σZW RXY V ξ −σY W e ∇ZσXV −σXV e ∇ZσY W +σY V e ∇ZσXW +σXW e ∇ZσY V for all X, Y, Z, V, W ∈Γ(Tf M). Using σuv = 0 we get (5.3) e ∇ze Rxyvw =∇zRxyvw for all z, x, y, v, w ∈Tmf M. Finally, taking covariant derivatives in (5.2) we obtain e ∇2 XX e RY ZY Z =∇2 XXRY ZY Z +σXX ∇ξRY ZY Z + 2 σ2 XY RξXξZ + 2 σ2 XZRξY ξY −4σXY σXZRξY ξZ + 2 σXY RTXZY Z + 2 σXZRY TXY Z + 2 e ∇XσXY RξZY Z +2 e ∇XσXZRY ξY Z −σY Y e ∇XXσZZ −σZZ e ∇XXσY Y +σY Z e ∇XXσY Z −2e ∇XσY Y e ∇XσZZ + 2 ³e ∇XσY Z´2+ 4σXY ∇XRξZY Z + 4σXZ∇XRY ξY Z, and using the expression for σand e ∇σat mwe get (5.4) e ∇2 xx e Ryzyz =∇2 xxRyzyz +2 3RxyxξRyzξz +2 3RxzxξRyzyξ −8 9RxyξyRxzξz +2 9R2 xyzξ +2 9R2 xzyξ +4 9RxyzξRxzyξ, for all x, y, z ∈Tmf M. Lemma 5.10 follows from (5.1), (5.3), (5.4) and the definitions of τ,kRk2,kρk2and ∆τafter doing some straightforward calculations. 106 5 Geodesic celestial spheres in Lorentzian manifolds Theorem 5.11. Let ¡Mn+1, g¢be a Lorentzian manifold and ξ∈TmMan instantaneous observer. The (n−1)–dimensional volume of the geodesic celestial spheres associated with ξ∈TmMsatisfies voln−1¡Gξ m(r)¢=cn−1rn−1µ1 + A(ξ) nr2+B(ξ) n(n+ 2)r4+O(r6)¶(m), where A(ξ) = −1 6(τ+ 2ρξξ), B(ξ) = −1 120 kRk2+1 45 kρk2+1 72 τ2−1 20 ∆τ−1 15 n X i,j,k=1 R2 ξijk +2 45 n X i,j=1 ρijRξiξj +1 18 n X i,j=1 R2 ξiξj +1 45 n X i=1 ρ2 ξi +1 30ρ2 ξξ +1 18τρξξ −1 10∆ρξξ −1 20∇2 ξξτ−1 10∇2 ξξρξξ. Proof. Since radial geodesics starting from morthogonally to ξare the same for Mand f M, it is clear that the geodesic celestial sphere Gξ m(r) of Massociated with the instantaneous observer ξ∈TmMcoincides with the geodesic sphere Gf M m(r) of radius rcentered at min the Riemannian manifold f Mfor sufficiently small radius. Now, the first terms in the power series expansion of the volume of sufficiently small geodesic spheres are well known [83]. This is also a special case of Theorem 4.20 for the Weyl invariant of degree 0, W= 1: vol ³Gf M m(r)´=cn−1rn−1½1−eτ 6nr2−r4 n(n+ 2)µke Rk2 120 −keρk2 45 −eτ2 72 +e ∆eτ 20 ¶+O(r6)¾(m). Using the relations in Lemma 5.10 the result follows. We also state here some algebraic preliminaries. Lemma 5.12. Let (V, h,i)a Lorentzian vector space and let Wdenote a covariant tensor of type (0,2k). If Wζ···ζ= 0 for all ζwith hζ, ζi=−1, then Wx···x= 0 for all x∈V. Proof. If ζis a timelike vector, we have, by hypothesis 0 = Wµζ p−hζ, ζi, . . . , ζ p−hζ, ζi¶=³−hζ, ζi´−kWζ···ζ, and thus Wζ···ζ= 0. Now, if xis an arbitrary vector, for sufficiently small ²,ζ+²x is timelike if ζis timelike. Then, 0 = W(ζ+²x, . . . , ζ +²x) = Wζ···ζ+···+²2kWx···x. Taking into account that ²is arbitrary, this immediately implies that Wx···x= 0 which proves the result. Lemma 5.13. Let ¡Mn+1, g¢be a Lorentzian manifold and let a,b,cbe real numbers with b6= 0. If a τ +b ρζζ =cat some point m∈Mfor all vector ζ∈TmMwith g(ζ, ζ) = −1, then the manifold is Einstein at m. 5.2.1 Power series expansions 107 Proof. The hypothesis can equivalently be written as −ag(ζ, ζ)τ+bρζζ +cg(ζ, ζ) = 0, which using Lemma 5.12 implies −ag(x, x)τ+bρxx +cg(x, x) = 0 for all x∈TmM. The result follows from linearity and symmetry of the Ricci tensor. For the purpose of analyzing the coefficient B(ξ) in Theorem 5.11, we define the following two tensors η(x, y) = n X i,j,k=0 ²i²j²kR(x, ei, ej, ek)R(y, ei, ej, ek), ω(x, y, v, w) = n X i,j=0 ²i²jR(x, ei, y, ej)R(v, ei, w, ej), where, as usual ²i=g(ei, ei) and x, y, v, w ∈TmM. Note that the definitions above are independent of the orthonormal basis chosen, and thus ωand ηare well defined tensors at a given point m∈M. We have the following result. Lemma 5.14. Let ¡Mn+1, g¢be an Einstein Lorentzian manifold. If there exist constants a, b, c, k ∈Rwith c6= 0 and 3c6= (n+ 5)bsuch that akRk2+b ηζζ +c ωζζζζ =k for all vectors ζ∈TmMwith g(ζ, ζ) = −1, then Mhas constant sectional curvature at m. Proof. Using Lemma 5.12, the hypothesis can be rewritten as (5.5) akRk2g(x, x)2−bg(x, x)η(x, x) + c ω(x, x, x, x) = kg(x, x)2 for all x∈TmM. For arbitrary α, β ∈Rand tangent vectors xand ywe get akRk2g2 αx+βy,αx+βy −b gαx+βy,αx+βyηαx+βy,αx+βy +c ωαx+βy,...,αx+βy =k g2 αx+βy,αx+βy. Since αand βare arbitrary, expanding the above equality and comparing the coefficients of α2β2we get, 2akRk2(gxxgyy + 2g2 xy)−b(gxxηyy + 4gxyηxy +gyyηxx) +2c(ωxxyy +ωxyxy +ωxyyx)=2k(gxxgyy + 2g2 xy). Setting y=eiin the above equality and contracting we have 2a(n+3) kRk2gxx −b(kRk2gxx +(n+5)ηxx)+cµ3ηxx +2 n X i,j=0 ²i²jρijRxixj¶= 2(n+3)k gxx. Since Mis Einstein, n X i,j=0 ²i²jρijRxixj =τ2 (n+ 1)2gxx, 108 5 Geodesic celestial spheres in Lorentzian manifolds and the above equation becomes (−b(n+5)+3c)ηxx =µ−2a(n+ 3) kRk2+bkRk2−2cτ2 (n+ 1)2+ 2k(n+ 3)¶gxx. Contracting again, (−b(n+ 5) + 3c)kRk2=µ−2a(n+ 3) kRk2+bkRk2−2cτ2 (n+ 1)2+ 2k(n+ 3)¶(n+ 1). Hence, ηxx =kRk2gxx/(n+ 1), which by the symmetry of ηand the metric tensor, it is equivalent to η=kRk2g/(n+ 1). As a consequence, using (5.5) we have (5.6) ωxxxx =1 cµ−akRk2+bkRk2 n+ 1 +k¶g(x, x)2. Next, we show that the above equation is an equivalent condition to constant sectional curvature for Lorentzian manifolds. Let π⊂TmMbe a plane of signature (−+) and let {ζ, ϑ}be an orthonormal basis of πwith g(ζ, ζ) = −g(ϑ, ϑ) = −1. The Jacobi operator Rζ(x) = R(ζ, x)ζis self–adjoint when restricted to ζ⊥, and thus it is diagonalizable with respect to an orthonormal basis {e1, . . . , en}of ζ⊥with eigenvalues λ1(ζ), . . . ,λn(ζ). Now, with respect to the orthonormal basis of TmM,{e0=ζ, e1, . . . , en}, Equation (5.6) gives n X i,j=1 R2 ζeiζej=1 cµb−(n+ 1)a n+ 1 kRk2+k¶. Hence, the the eigenvalues λα(ζ) are bounded independently of the timelike unit ζbecause λα(ζ)2=R2 ζeαζeα≤ n X i,j=1 R2 ζeiζej=1 cµb−(n+ 1)a n+ 1 kRk2+k¶, for all α∈ {1, . . . , n}. Writing ϑ=Pn i=1 ϑieiwith respect to the basis above, one has that the sectional curvature of πsatisfies K(π) = −Rζϑζϑ =− n X i,j=1 ϑiϑjRζeiζej=− n X i=1 (ϑi)2λi(ζ). Since hϑ, ϑi= 1 = Pn i=1(ϑi)2, one has |K(π)| ≤ Pn i=1(ϑi)2|λi(ζ)| ≤ Cfor some constant C. This shows that the sectional curvature is bounded on planes of signature (+−) and therefore, Mhas constant curvature at m(see [7], [96], [108]). Remark 5.15.Equation (5.6) is equivalent to the 2–stein condition. See [79] for a different proof that 2–stein Lorentzian manifolds have constant curvature. 5.2.2 Volume comparison theorems 109 5.2.2 Volume comparison theorems It is well known that the sectional curvature of a semi–Riemannian manifold is bounded from above or from below if and only if it is constant [7], [108]. In order to derive volume comparison theorems we need a revision of the boundedness conditions on the sectional curvature. It seems natural to impose such curvature bounds on the curvature tensor itself rather than on the sectional curvature. Following [5], we write R≥λor R≤λif and only if for all x, y ∈TM R(x, y, x, y)≥λ(g(x, x)g(y, y)−g(x, y)2) or R(x, y, x, y)≤λ(g(x, x)g(y, y)−g(x, y)2), respectively. Note that the first condition above (resp. the second condition) is equivalent to requiring the sectional curvature to be bounded from below (resp. from above) on planes of signature (++) and from above (resp. from below) on planes of signature (+−). Examples of semi–Riemannian manifolds whose curvature tensor is bounded as above can easily be produced as follows: •Let (M1, g1), (M2, g2) be Riemannian manifolds with non–negative KM1≥0 and non–positive KM2≤0 sectional curvature, respectively. Then the product manifold (M1×M2, g1−g2) is a semi–Riemannian manifold whose curvature tensor satisfies R≥0. See [5] for related examples. •A more general construction of Lorentzian manifolds with bounded curvature is as follows. Let (M, g) be a conformally flat Lorentz manifold whose Ricci tensor is diagonalizable, ρ=diag(µ0, µ1, . . . , µn), where the distinguished eigenvalue µ0corresponds to a timelike eigenspace. If µ0≥max{µ1, . . . , µn}(resp. µ0≤min{µ1, . . . , µn}) then R≤λ(resp. R≥λ) for some constant λ. Note that the previous construction applies to Robertson–Walker space–times as well as to locally conformally flat static space–times whose rest–spaces are of constant sectional curvature [24]. Although it is not possible to obtain direct information of the Ricci tensor from the boundedness conditions above, an important observation for the purpose of studying volume properties of geodesic celestial spheres is the following. Let ξbe an instantaneous observer at m∈Mand complete it to an orthonormal basis {e0=ξ, e1, . . . , en}of TmM. Then τ+ 2ρξξ =Pn i,j=1 Rijij. Hence by assuming R≥λ(resp. R≤λ), we have τ+ 2ρξξ ≥n(n−1)λ(resp. τ+ 2ρξξ ≤n(n−1)λ). We are now ready to prove a Bishop–G¨unther type theorem [43], [46]. Theorem 5.16. Let (Mn+1, g)be a (n+1)–dimensional Lorentzian manifold and Nn+1(λ) a Lorentzian manifold of constant sectional curvature λ. The following statements hold: (i) If R≥λ, then volM n−1¡Gξ m(r)¢≤volN(λ) n−1(G(r)) , for all sufficiently small rand all instantaneous observers ξ∈TmM. 110 5 Geodesic celestial spheres in Lorentzian manifolds (ii) If R≤λ, then volM n−1¡Gξ m(r)¢≥volN(λ) n−1(G(r)) , for all sufficiently small rand all instantaneous observers ξ∈TmM. Moreover, the equality holds in (i) or (ii) for all ξ∈TmMif and only if Mhas constant sectional curvature λat m. Proof. Assume that R≥λ. If R≤λthe result is obtained in a similar way. As usual, let {e0=ξ, e1, . . . , en}be an orthonormal basis of TmM. As we have already seen, τ+ 2ρξξ = Pn i,j=1 Rijij. Hence, τ+ 2ρξξ ≥n(n−1)λ. Thus, by Theorems 5.9 and 5.11, we have for sufficiently small r volM n−1¡Gξ m(r)¢=cn−1rn−1µ1−τ+ 2ρξξ 6nr2+O(r4)¶ ≤cn−1rn−1µ1−n−1 6λ r2+O(r4)¶= volN(λ) n−1(G(r)) , which proves the first part of the assertion. Now, assume that the equality holds for sufficiently small rand all ξ∈TmM. Then, τ+ 2ρξξ =n(n−1)λfor all ξ∈TmM. We prove that this implies that the sectional curvature Kis constant K=λon planes of signature (++). Given πa plane of signature (++), we take an orthonormal basis {x, y}of πand we complete it to an orthonormal basis {e0, e1=x, e2=y, . . . , en}of TmMwith e0timelike. Then n X i,j=1 Rijij =τ+ 2ρe0e0=n(n−1)λ and since Rijij ≥λby assumption, it follows that K(π) = λ. Now the constancy of the sectional curvature at mfollows from [108]. The previous theorem shows that volM n−1¡Gξ m(r)¢/volN(λ) n−1(G(r)) ≤1 (resp. ≥1) if R≥λ(resp. R≤λ). A more precise result in the spirit of Gromov’s theorem can be stated as follows [43], [46] Theorem 5.17. Let (Mn+1, g)be a (n+1)–dimensional Lorentzian manifold and Nn+1(λ) a Lorentzian manifold of constant sectional curvature λ. (i) If R≥λ, then r7→ volM n−1¡Gξ m(r)¢ volN(λ) n−1(G(r)) is non–increasing for sufficiently small rand all instantaneous observers ξ∈TmM. 5.2.3 Characterization of locally isotropic Lorentzian manifolds 111 (ii) If R≤λ, then r7→ volM n−1¡Gξ m(r)¢ volN(λ) n−1(G(r)) is non–decreasing for sufficiently small rand all instantaneous observers ξ∈TmM. Proof. By using the results in Theorem 5.9 and Theorem 5.11 one gets the first terms in the power series expansion of the quotient volM n−1¡Sξ m(r)¢ volN(λ) n−1(S(r)) = 1 + µn(n−1)λ−(τ+ 2ρξξ) 6n¶r2+O(r4). Therefore, if R > λ, we have τ+ 2ρξξ > n(n−1)λ. Hence the derivative of the quotient is negative for small r, and thus the quotient is decreasing, which shows (i), since in case R=λthe quotient above is constant for sufficiently small r. The proof of (ii) is completely analogous. Remark 5.18.Under the hypothesis of Theorem 5.17, if there exists 0 < r0< r1such that volM n−1¡Gξ m(r0)¢ volN(λ) n−1(G(r0)) =volM n−1¡Gξ m(r1)¢ volN(λ) n−1(G(r1)) , then the sectional curvature is constant. Indeed, since the quotient above is monotone, then it must be constant and thus R=λ(see proof of Theorem 5.17). Remark 5.19.We point out here that the proofs of Theorem 5.16 and 5.17 only require the boundedness conditions to hold for spacelike planes. 5.2.3 Characterization of locally isotropic Lorentzian manifolds We recall that a Lorentzian manifold is said to be locally isotropic if for each point m∈M and all pair of non–null vectors x, y ∈TmMwith g(x, x) = g(y, y) there exists a local isometry of (M, g) fixing mand transforming xinto y. If Mis locally isotropic, volM n−1¡Gξ m(r)¢does not depend on the instantaneous observer ξ∈TmM. Moreover, since locally isotropic Lorentzian manifolds are locally homogeneous, it follows that volM n−1¡Gξ m(r)¢does not depend on the center m. The following theorem shows that local isotropy can be recovered from the properties of the volume of geodesic celestial spheres [46]. Theorem 5.20. Let ¡Mn+1, g¢be a Lorentzian manifold. If the volume of the geodesic celestial spheres Gξ m(r)is independent of the observer field ξ∈TM, then Mhas constant sectional curvature. Proof. If the volume of each geodesic celestial sphere Gξ m(r) is independent of the instantaneous observer ξ∈TmM, then the coefficients A(ξ) and B(ξ) in the power series expansion of voln−1(Gξ m(r)) in Theorem 5.11 are independent of ξ. Now, as −(τ+ 2ρξξ)/6 = A(ξ) is 112 5 Geodesic celestial spheres in Lorentzian manifolds constant, using Lemma 5.13, one has that that Mis Einstein, and thus ρ=τ n+1g. Hence, it follows from the second coefficient B(ξ) in Theorem 5.11 that constant =B(ξ) = −1 120 kRk2+5n2+ 38n+ 61 360(n+ 1)2τ2+1 18 n X i,j=1 R2 ξiξj −1 15 n X i,j,k=1 R2 ξijk. With the notation of Lemma 5.14 we have ωξξξξ = n X i,j=0 ²i²jR2 ξiξj = n X i,j=1 R2 ξiξj and ηξξ = n X i,j,k=0 ²i²j²kR2 ξijk = n X i,j,k=1 R2 ξijk −2 n X i,j=1 R2 ξiξj, and it follows from that lemma that the sectional curvature of Mis constant. Corollary 5.21. Let ¡Mn+1, g¢be a Lorentzian manifold. If the volume of each geodesic celestial sphere of sufficiently small radius coincides with the corresponding one of a geodesic celestial sphere of the same radius in a space of constant sectional curvature λ, then M has constant sectional curvature λ. Proof. Theorem 5.9 implies that in a Lorentzian manifold of constant sectional curvature the volume of geodesic celestial spheres does not depend on the instantaneous observer. Using Theorem 5.20 we deduce that Mhas constant sectional curvature λ. Doing the power series expansion of the formula in Theorem 5.9 we get voln−1¡GN(λ)(r)¢=cn−1rn−1½1−n−1 6λ r2+O¡r4¢¾. Comparing the coefficient of r2in the above power series expansion with the corresponding one in the formula of Theorem 5.11 we get that the sectional curvature is exactly λ. Since the concept of local isotropy is equivalent to constant sectional curvature for Lorentzian manifolds, the results of this section can be condensed for n+ 1 ≥3 in the following Theorem 5.22. A Lorentzian manifold is locally isotropic if and only if the volume of its geodesic celestial spheres is independent of the instantaneous observer. 5.3 Total curvatures of geodesic celestial spheres The main purpose of this section is to investigate the curvature of geodesic celestial spheres by focusing on the properties of their total curvatures associated with simple Weyl invariants. We show that Lorentzian manifolds of constant sectional curvature can be characterized by means of total curvatures of geodesic celestial spheres (see Theorem 5.26). 5.3 Total curvatures of geodesic celestial spheres 113 From now on, all geometric objects defined on a geodesic celestial sphere will be denoted using the symbol ˆ. Let Wbe a simple Weyl invariant. We define the total scalar curvature of the geodesic celestial sphere Gξ m(r)associated with the simple Weyl invariant Was [42] Wm(ξ, r) = ZGξ m(r) ˆ W. As stated before, ˆ Wdenotes the corresponding simple Weyl invariant in the geodesic celestial sphere Gξ m(r). As it was the case for the volume of geodesic celestial spheres, it is clear from the definition that the total scalar curvature associated with a simple Weyl invariant depends on the radius r, the base point m, the instantaneous observer ξ∈TmMand the Weyl invariant Winvolved in its construction. If the manifold has constant sectional curvature there is no dependence on the point or the instantaneous observer. In fact, an exact formula may be obtained. Theorem 5.23. Let ¡Mn+1, g¢a Lorentzian manifold of constant sectional curvature λ. For each point m∈Mand each instantaneous observer ξ∈TmMthe total scalar curvature Wm(ξ, r)associated with the simple Weyl invariant Wof degree 2νis voln−1¡Gξ m(r)¢=                cn−1(n−1)(n−2)AW(n−1)µsin r√λ √λ¶n−1−2ν , λ > 0, cn−1(n−1)(n−2)AW(n−1)rn−1−2ν, λ = 0, cn−1(n−1)(n−2)AW(n−1)µsinh r√−λ √−λ¶n−1−2ν , λ < 0, where AWis the polynomial given in Remark 4.2. Proof. Since the manifold Mhas constant curvature, the submanifold f M= expm¡U∩ξ⊥¢ defined in the previous section has also sectional curvature λ. The geodesic celestial sphere Gξ m(r) is the geodesic sphere of radius rcentered at mof the n–dimensional Riemannian submanifold f M. Then, the total scalar curvature of the geodesic celestial sphere Gξ m(r) associated with Wis the total scalar curvature of the geodesic sphere Gm(r) of f Massociated with W. The latter was given in Example 4.17 for positive curvature. For zero and negative curvature we get analogous expressions, the ones appearing in the statement of this theorem. In order to derive a characterization of locally isotropic Lorentzian manifolds we need the first terms of the power series expansion of r7→ Wm(ξ, r). These are calculated in what follows. We assume that {e0=ξ, e1, . . . , en}is an orthonormal basis of TmM. 120 Part III Real hypersurfaces in the complex hyperbolic space 121 The aim of submanifold geometry is to understand geometric invariants of submanifolds and to classify submanifolds according to given geometric data. In Riemannian geometry, the structure of a submanifold is encoded in the second fundamental form and its geometry is controlled by the equations of Gauss, Codazzi and Ricci. The situation simplifies for hypersurfaces, as the Ricci equation is trivial and the second fundamental form can be written in terms of a self–adjoint tensor field, the shape operator. The eigenvalues of the shape operator, the so–called principal curvatures, are the simplest geometric invariants of a hypersurface. Two basic problems in submanifold geometry are to understand the geometry of hypersurfaces for which the principal curvatures are constant, and to classify them. This problem has a long history and over the years many surprising features have been discovered. ´ E. Cartan [28] showed that in spaces of constant curvature a hypersurface has constant principal curvatures if and only if it is isoparametric. There is a remarkable interplay between the geometry and topology of isoparametric hypersurfaces in spheres Sn. Using methods from algebraic topology, H. F. M¨unzner [101] proved that the number gof distinct principal curvatures of an isoparametric hypersurface in Snis 1, 2, 3, 4 or 6. In a series of papers, [28], [29], [30], [31], ´ E. Cartan investigated isoparametric hypersurfaces in spheres and classified those for which gis at most three. It follows from his work that all isoparametric hypersurfaces in spheres with g≤3 are open parts of homogeneous hypersurfaces. It is obvious that homogeneous hypersurfaces have constant principal curvatures. The homogeneous hypersurfaces in spheres have been classified by W.–Y. Hsiang and H. B. Lawson [89]. It follows from this classification that homogeneous hypersurfaces with g= 6 exist only in spheres of dimension 7 and 13. U. Abresch [1] then proved that isoparametric hypersurfaces with g= 6 exist only in S7and S13. This naturally leads to the conjecture that any isoparametric hypersurface in a sphere with g= 6 is an open part of a homogeneous hypersurface. This was answered affirmatively by J. Dorfmeister and E. Neher [51] for n= 7, but for n= 13 the problem is still open. Surprisingly, for g= 4 there are inhomogeneous isoparametric hypersurfaces. The first such examples were constructed by H. Ozeki and M. Takeuchi [110]. D. Ferus, H. Karcher and H.–F. M¨unzner 123 [58] then constructed series of inhomogeneous isoparametric hypersurfaces in spheres using representations of real Clifford algebras. A remarkable result by S. Stolz [121] says that the principal curvatures and their multiplicities of any isoparametric hypersurface with g= 4 in a sphere coincide with the ones of either the homogeneous hypersurfaces or the hypersurfaces constructed by D. Ferus, H. Karcher and H.–F. M¨unzner. T. Cecil, Q.–S. Chi and G. Jensen [32] recently proved that with 10 possible exceptions all isoparametric hypersurfaces in spheres with g= 4 are among the known homogeneous or inhomogeneous examples. Whereas the classification problem of isoparametric hypersurfaces in spheres is rather involved, it is much simpler in its non–compact dual, the real hyperbolic space RHn. In fact, using the Gauss and Codazzi equations, ´ E Cartan [28] showed that the number g of distinct principal curvatures of an isoparametric hypersurface in RHnis either 1 or 2. This easily leads to a complete classification: geodesic hyperspheres, horospheres, totally geodesic hyperplanes and its equidistant hypersurfaces, tubes around totally geodesic subspaces of dimension greater or equal than one. As a consequence, all isoparametric hypersurfaces in real hyperbolic spaces are open parts of homogeneous hypersurfaces. The isoparametric hypersurfaces in Euclidean spaces were classified by T. Levi–Civita [97] for dimension 3 and by B. Segre [119] for arbitrary dimensions. Also here all isoparametric hypersurfaces are open parts of homogeneous hypersurfaces. In complex space forms the notions of isoparametric hypersurfaces and hypersurfaces with constant principal curvatures are not the same [127]. In fact, Q.–M. Wang [130] gave an example of an isoparametric hypersurface in complex projective space CPnwith non–constant principal curvatures. The current state of the classification problem of hypersurfaces with constant principal curvatures in complex space forms is as follows. We continue to denote by gthe number of distinct principal curvatures. To emphasize that the real codimension of the hypersurface is one (and not two as it is for a complex hypersurface) we use the notion of a real hypersurface. Y. Tashiro and S. I. Tachibana [126] proved that there are no totally umbilical real hypersurfaces in non–flat complex space forms. Thus the case g= 1 cannot occur. If ξis a (local) unit normal field of a real hypersurface Min a complex space form ¯ M, and Jdenotes the complex structure of ¯ M, then Jξ is tangent to Meverywhere. The vector field Jξ is called the Hopf vector field on M. The hypersurface Mis said to be a Hopf hypersurface if Jξ is a principal curvature vector of Meverywhere. We assume n≥2. Using the classification of homogeneous hypersurfaces in spheres and the Hopf map S2n+1 →CPn, R. Takagi [123] derived the classification of homogeneous real hypersurfaces in complex projective spaces. All of them are Hopf hypersurfaces, and the number gof distinct principal curvatures is either 2, 3 or 5. R. Takagi then proved in [124] and [125] that every real hypersurface with two or three distinct constant principal curvatures in CPnis an open part of a homogeneous hypersurface. The case g= 3 and n= 2 was omitted by R. Takagi and settled later by Q.-M. Wang [131]. M. Kimura [91] showed that every Hopf hypersurface in CPnwith constant principal curvatures is an open part of a homogeneous hypersurface in CPn. It is still unknown whether for any real hypersurface with constant principal curvatures in CPnthe number gis necessarily 2, 3 or 5. Also, 124 there is no known example of a real hypersurface with constant principal curvatures in CPnwhich is not an open part of a homogeneous real hypersurface. S. Montiel [99] proved that every real hypersurface with two distinct constant principal curvatures in complex hyperbolic space CHn,n≥3, is an open part of a geodesic sphere, of a horosphere, of a tube around a totally geodesic CHn−1⊂CHn, or of a tube with radius log(2+√3) around a totally geodesic RHn⊂CHn. All these real hypersurfaces are homogeneous Hopf hypersurfaces. For n= 2 this problem is still open. J. Berndt derived in [10] the classification of all Hopf hypersurfaces with constant principal curvatures in CHn. Any such hypersurface is an open part of a horosphere, of a tube around a totally geodesic CHk⊂CHnfor some k∈ {0, . . . , n −1}, or of a tube around a totally geodesic RHn⊂CHn. All these tubes and horospheres are homogeneous hypersurfaces. This naturally leads to the question whether all homogeneous real hypersurfaces in CHnare necessarily Hopf hypersurfaces. The answer to this question is negative. In [11] J. Berndt constructed homogeneous hypersurfaces in CHnwhich are not Hopf hypersurfaces. J. Berndt and M. Br¨uck constructed in [12] new examples of homogeneous real hypersurfaces in CHn. J. Berndt and Tamaru [16] showed recently that these new examples, together with the above mentioned homogeneous real hypersurfaces, provide the complete classification of homogeneous real hypersurfaces in CHn. The number of distinct principal curvatures of all these homogeneous real hypersurfaces is either 2, 3, 4 or 5. No examples are known of real hypersurfaces with constant principal curvatures in CHnwhich are not an open part of a homogeneous real hypersurface. It is also not known whether for any real hypersurface with constant principal curvatures in CHnthe number gof distinct principal curvatures must necessarily be 2, 3, 4 or 5. In Chapter 6 we study cohomogeneity one actions on the complex hyperbolic space. Based on the classification given by J. Berndt and H. Tamaru, we focus on the geometry of the orbits of the cohomogeneity one actions described in [16]. This is accomplished in Section 6.3. In Chapter 7 we carry out the classification of real hypersurfaces in CHn with three distinct constant principal curvatures. In particular, our result implies that real hypersurfaces in complex hyperbolic spaces with at most three constant principal curvatures are homogeneous submanifolds. 125 126 Chapter 6 Cohomogeneity one actions on the complex hyperbolic space In this chapter we study the geometry of the orbits of a cohomogeneity one action on CHn [15]. In Section 6.1 we give the basic definitions and concepts needed to describe cohomogeneity one actions. We explain some facts of cohomogeneity one actions on Hadamard manifolds and give an overview of the situation in Rnand RHn. Then, Section 6.2 is devoted to presenting a suitable description of CHn. The conventions and results explained throughout this section are used in the rest of the chapter, sometimes without explicit mention to them. Finally, Section 6.3 carries out the study of cohomogeneity one actions on the complex hyperbolic space with special attention to the description of the singular orbits of cohomogeneity one actions with one non–totally geodesic singular orbit. In particular we emphasize Theorems 6.8 and 6.16 as they are used in the following chapter. 6.1 Preliminaries Let Mbe a Riemannian manifold and Ga Lie group. A G–action on Mor an action of G on Mis a map G×M−→ M (g, p)7→ gp such that ep =pfor all p∈M, where eis the identity of G, and g(hp)=(gh)pfor all g, h ∈Gand p∈M. If p∈M, then G·p={gp :g∈G}is the orbit of Gthrough pand Gp={g∈G:gp =p}is the isotropy group of Gat p. If M=G·pfor some p∈M, then the action of Gis said to be transitive and Mis called a homogeneous G– space. Homogeneous spaces are of great interest in differential geometry. See [88] for a comprehensive introduction to the subject. An isometric action of Gon Mis a G–action such that for any fixed g∈G, the map p7→ gp is an isometry of M. From now on, we assume that Gis a connected closed subgroup of the isometry group of Macting on Min the usual way. 127 128 6 Cohomogeneity one actions on the complex hyperbolic space We denote by M/G the set of orbits of the action of Gon Mand equip M/G with the quotient topology relative to the canonical projection p∈M→G·p∈M/G. Since Gis a closed subgroup of the isometry group of M, the quotient space M/G is a Hausdorff space and each orbit G·pis a closed embedded submanifold [90]. Moreover, Gpis compact, G·p is a Riemannian homogeneous space G·p=G/Gpand Gacts transitively on G·pby isometries. D. Montgomery and C. T. Yang introduced in [98] the concept of a slice. This notion provides the technical machinery which allows us to define a partial ordering on the set of orbit types. We say that two orbits G·pand G·qhave the same orbit type if Gpand Gqare conjugate in G. This defines an equivalence relation among the orbits of G. We denote by [G·p] the corresponding equivalence class of G·pand we call [G·p] the orbit type of G·p. We introduce a partial ordering on the moduli space of orbit types. We put [G·p]≤[G·q] if and only if Gqis conjugate in Gto some subgroup of Gp. There exists a largest orbit type in the moduli space of orbit types. Each representative of this largest orbit type is called a principal orbit. The union of all principal orbits forms a dense and open subset of M. Each principal orbit is an orbit of maximal dimension. A non–principal orbit with the same dimension as a principal orbit is called an exceptional orbit. An orbit whose dimension is less that the dimension of a principal orbit is called a singular orbit. Acohomogeneity one action of Gon a manifold Mis an isometric action of Gon M such that the codimension of each principal orbit is one. We say that two cohomogeneity one actions are orbit equivalent if there is an isometry of Mthat maps the orbits of one action onto the orbits of the other action. An embedded submanifold of a Riemannian manifold Mis said to be extrinsically homogeneous, if there exists an isometry of Mthat acts transitively on the submanifold and leaves it invariant. Cohomogeneity one actions are intimately related to extrinsically homogeneous hypersurfaces. Indeed the classification problem of cohomogeneity one actions up to orbit equivalence is equivalent to the classification of extrinsically homogeneous hypersurfaces up to isometry congruence. P. S. Mostert [100] and L. B´erard Bergery [8] proved that the orbit space M/G of a cohomogeneity one action is homeomorphic to R,S1, [0,1] or [0,∞). This result implies that a cohomogeneity one action has at most two singular or exceptional orbits corresponding to the boundary points of M/G. If there exists one singular orbit, each principal orbit is geometrically a tube around the singular orbit. If there are no singular or exceptional orbits, in which case M/G is homeomorphic either to Ror S1, the orbits of the action of Gon Mform a Riemannian foliation on M. Moreover, since principal orbits are always homeomorphic to each other, the projection M→M/G is a fiber bundle. Assume Gis a connected closed subgroup of the isometry group of Macting on M with cohomogeneity one. Let Fbe a singular or exceptional orbit of the action. Then, the isotropy group Gpat p∈Facts transitively on the unit sphere of the normal space of F at p. This implies that any singular or exceptional orbit of a cohomogeneity one action is minimal [12]. Moreover, if dim(G·p)<(dim M−1)/2 then G·pis totally geodesic in M [112]. From now on we assume that Mis a Hadamard manifold, that is, a connected, simply 6.1 Preliminaries 129 connected, complete Riemannian manifold of non–positive curvature. As Mis simply connected, M/G cannot be homeomorphic to S1. This follows from the exact homotopy sequence of a fiber bundle with connected fibers and base space S1 ··· → π1(M)→π1(M/G)→π0(F)→ ··· . where Fis the fiber. A cohomogeneity one action on a Hadamard manifold cannot have exceptional orbits and it can have at most one singular orbit. Therefore, M/G is homeomorphic to Ror to [0,∞). The above assertions can be improved in the following way (see [12] and [111]). Theorem 6.1. Let Gbe a connected closed subgroup of the isometry group of an n– dimensional Hadamard manifold Macting on Mwith cohomogeneity one. Then one of the following two possibilities holds: (a) All orbits are principal and the isotropy group at any point is a maximal compact subgroup of G. Any orbit is diffeomorphic to Rn−1and there exists a solvable connected closed subgroup of Gacting simply transitively on each orbit. (b) There exists exactly one singular orbit Fand the isotropy group at any point of F is a maximal compact subgroup of G. The singular orbit is diffeomorphic to Rkfor some k∈ {0, . . . , n −2}and there exists a solvable connected closed subgroup of G acting simply transitively on F. Any principal orbit is a tube around Fand thus diffeomorphic to Rk×Sn−k−1. Among all Hadamard manifolds, of special interest are the Euclidean space and all rank one symmetric spaces of non–compact type. The cohomogeneity one actions on the Euclidean space were classified by T. Levi–Civita [97] and B. Segre [119]. Theorem 6.2. Let Gbe a Lie subgroup of the isometry group of Rn,Rn×τO(n), acting on Rnwith cohomogeneity one. Then the action of Gis orbit equivalent to one of the following actions: (i) The action of SO(n)⊂Rn×τO(n). The singular orbit is a point and the principal orbits are spheres. (ii) The action of Rk×τSO(n−k)⊂Rn×τO(n)for some k∈ {1, . . . , n −2}. There is one singular orbit which is a totally geodesic Rk⊂Rnand the principal orbits are tubes around it. (iii) The action of Rn−1⊂Rn×τO(n). All orbits are principal and totally geodesic hyperplanes. The classification of cohomogeneity one actions on the real hyperbolic space follows from the work by ´ E. Cartan [28], where he classified all the hypersurfaces with constant principal curvatures in the real hyperbolic space RHn. Every principal orbit of a cohomogeneity one action has constant principal curvatures. Hence, Cartan’s result applies and we get 136 6 Cohomogeneity one actions on the complex hyperbolic space Furthermore, Tαand Tβare complex distributions on Mand the Hopf vector field is a principal curvature vector of γat any point of M. Thus, Mis a Hopf hypersurface. More specifically, let p∈Mand let cbe the geodesic of CHndefined by the initial condition c0(0) = ξp. Then c(r) is a point in the totally geodesic CHk. The principal curvature vector subspace Tα(p) is the parallel translate of Tc(r)CHkalong the geodesic cand Tβ(p) is the parallel translate of T⊥ c(r)CHkªRJc0(r) along the geodesic c. In the above discussion if k= 0, that is, the singular orbit is a point, then the principal orbits are geodesic spheres and there are just two eigenvalues α=1 2tanh r 2and γ= coth r with multiplicities 2(n−1) and 1 respectively. Similarly, if k=n−1 then the singular orbit is a totally geodesic CHn−1⊂CHn and the tubes around it have only two constant principal curvatures β=1 2coth r 2and γ= coth rwith corresponding multiplicities 2(n−1) and 1. The action of SO0(1, n) The group G=SO0(1, n)⊂SU(1, n) acts on CHnwith cohomogeneity one. This action has one singular orbit which is a totally geodesic RHn⊂CHn. The second fundamental form is completely determined by II = 0. Such a totally geodesic RHncan be constructed as follows. Let o∈CHnand choose V⊂ToCHna real linear subspace of the tangent space of real dimension n. Then, expo(V) is a totally geodesic RHn. We briefly discuss the geometry of the principal orbits of the action of G. Let Mbe one of these orbits. Then, Mis a tube of certain radius r > 0 around the singular orbit. Using standard Jacobi vector field theory we get that Mhas three principal curvatures. We choose the outward unit normal vector field ξso that the principal curvatures with respect to it are positive. The three principal curvatures are α=1 2tanh r 2, β =1 2coth r 2, γ = tanh r, with corresponding multiplicities mα=n−1, mβ=n−1, mγ= 1. The distributions Tαand Tβare real and the Hopf vector field is a principal curvature vector of γ. Thus, Mis a Hopf hypersurface. Let p∈Mand let cbe the geodesic of CHndefined by the initial condition c0(0) = ξp. Then c(r) is a point in the totally geodesic RHn. The principal curvature vector subspace Tα(p) is the parallel translate of Tc(r)RHnªRJc0(r) along the geodesic cand the principal vector subspace Tβ(p) is the parallel translate of T⊥ c(r)RHnªRc0(r) along the geodesic c. A special situation occurs when r= log(2 + √3). In this case β=γand there are just two principal curvatures αand γwith multiplicities n−1 and n. Both Tα(p) and Tγ(p) keep being real and the Hopf vector field is a principal vector field. 6.3.2 Cohomogeneity one actions with no singular orbits 137 6.3.2 Cohomogeneity one actions with no singular orbits Cases (iii) and (iv) in Theorem 6.4 correspond to cohomogeneity one actions on the complex hyperbolic space with no singular orbits. These cohomogeneity one actions arise naturally from the Iwasawa decomposition of SU(1, n). Different choices for the Iwasawa decomposition lead to congruent actions. The horosphere foliation Let KAN be an Iwasawa decomposition of the isometry group of CHnwith respect to some point o∈CHnand some point at infinity x∈CHn(∞). The group Nacts on CHnwith cohomogeneity one and all the orbits are principal. The resulting foliation is the well–known horosphere foliation. It contains the Heisenberg group Nas a horosphere and any other orbit is a suitable left translate of it. This foliation is constructed in the following way. Let cbe a unit speed geodesic with c(0) = o. We define the Busemann function,Bc:CHn→R, with respect to cas Bc(p) = lim t→∞(d(p, c(t)) −t), where dstands for the Riemannian distance function. The level sets of this function are called horospheres. A horosphere has the following geometrical interpretation. Consider the geodesic sphere centered at c(r) of radius r. This geodesic sphere contains o. In the complex hyperbolic space such geodesic spheres are defined for any r > 0. The limit set of these geodesic spheres when rtends to infinity is a horosphere. Different choices of oalong cgive all the different horospheres of the horosphere foliation determined by the point at infinity x= limt→∞ c(t). A horosphere has exactly two distinct principal curvatures α=1 2and β= 1, with corresponding multiplicities mα= 2(n−1) and mβ= 1. As usual, we choose the unit normal vector ξso that the principal curvatures are positive. The principal vector space of αis the orthogonal complement of the complex span of the unit normal vector ξ. Hence, Tαis a complex distribution. The Hopf vector field Jξ is a principal curvature vector of the principal curvature β. Hence, every horosphere is a Hopf hypersurface with constant principal curvatures. We have the following rigidity result. The proof is an easy consequence of Theorem 6.5. However, we sketch the proof as it was given in [10] because of its geometric interest. Theorem 6.7 (Rigidity of horospheres in CHn). Let Mbe a real hypersurface of the complex hyperbolic space with principal curvatures 1/2and 1. Then Mis an open part of a horosphere. 138 6 Cohomogeneity one actions on the complex hyperbolic space Sketch of the proof. Let ξdenote a local unit vector of M. Using the Gauss and Codazzi equations one can show that Jξ is a principal vector associated with the eigenvalue 1. For any p, let cpbe the geodesic cp(t) = expp(tξp). For r≥0 let Φrbe the map defined by Φr(p) = expp(rξp). Let v∈TpM. Using Jacobi vector field theory one gets Φr∗(v) = e−r/2Bv(r) if v∈T1/2(p) and Φr∗(v) = e−rBv(r) if v∈RJξp. This implies kΦr∗(v)k ≤ e−r/2kvkfor all v∈TM. Let ddenote the Riemannian distance function of CHnand dM the Riemannian distance function of M. The above inequality shows that d(cp(r), cq(r)) ≤ e−r/2dM(p, q) for any p, q ∈Mand r≥0. Using the triangle inequality we get d(p, co(t)) −t=d(p, co(t)) −d(p, cp(t)) ≤d(cp(t), co(t)) ≤e−t/2dM(p, o). Then the Busemann function verifies Bc(p) = 0 for all p∈M, which proves that Mis an open part of a horosphere. The solvable foliation As usual, let a⊕z⊕vbe the Lie algebra of the solvable part of the Iwasawa decomposition KAN with respect to some o∈CHnand x∈CHn(∞). Let us take wa linear hyperplane in v. Then h=a⊕z⊕wis a Lie subalgebra of a⊕z⊕vof codimension one. If H is the connected, simply connected Lie subgroup whose Lie algebra is h, then Hacts on CHnwith cohomogeneity one. The resulting cohomogeneity one action has no singular orbits and therefore induces a foliation on CHn. We call it the solvable foliation of CHn. Different choices of wlead to congruent actions. We mainly follow [11]. The orbit H·othrough ois the unique minimal orbit of this action. We study its geometry in more detail. The maximal complex subspace of c⊂his a Lie subalgebra and the connected, simply connected Lie subgroup whose Lie algebra is this maximal complex subspace cacts on H·oby left translation. The resulting orbit through ois a totally geodesic CHn−1⊂CHn. Indeed, H·ois ruled by totally geodesic CHn−1in CHn. The orthogonal complement hªcis one–dimensional and induces in H·oan integrable distribution Dby left translation of hªc. Each integral curve of Dthrough p∈H·ois a horocycle in the totally geodesic RH2determined by Dpand x∈CHn(∞). By definition we denote by W2n−1 a manifold constructed in this way. As we stated before all W2n−1are holomorphically congruent to each other. In Subsection 6.3.3 we generalize this construction and give more details about it. For the moment we content ourselves with the present description and study the geometry of the other orbits. Any other orbit of the action of His an equidistant hypersurface to this minimal one. Any two such orbits are congruent to each other if and only if their distance to H·ois the same. None of them is ruled by a totally geodesic CHn−1in the above sense. Let Mdenote an orbit of Hat a distance r≥0 from H·o. If r= 0 we consider the orbit H·oitself. The shape operator Sof Mhas exactly three eigenvalues α=1 2tanh r 2, β =3 4tanh r 2−1 2r1−3 4tanh2r 2, γ =3 4tanh r 2+1 2r1−3 4tanh2r 2. 6.3.2 Cohomogeneity one actions with no singular orbits 139 with corresponding multiplicities mα= 2n−3, mβ= 1, mγ= 1. The principal vector space of αis neither real nor complex. If ξdenotes the unit normal of M, the Hopf vector field Jξ is not a principal vector field and hence Mis not a Hopf hypersurface. Indeed, Jξ has non–trivial orthogonal projections onto Tβand Tγ. For the orbit H·owe have r= 0 and the principal curvatures become 0, −1/2 and 1/2 with multiplicities 2n−3, 1 and 1. We clearly see then that H·ois minimal. The following theorem shows that this eigenvalue structure is characteristic of this orbit [14]. It is a consequence of Theorem 6.16 which we prove later in Section 6.3.3. We also use some elementary results of the following chapter which we avoid repeating here to focus our attention on the main argument. Theorem 6.8 (Rigidity of the submanifold W2n−1). Let Mbe a connected real hypersurface in CHn,n≥3, with three distinct principal curvatures 0,−1/2and 1/2and multiplicities 2n−3,1and 1, respectively. Then Mis holomorphically congruent to an open part of the ruled real hypersurface W2n−1. Proof. Let ξbe the corresponding unit normal vector of M. Let p∈Mand suppose that the orthogonal projection of Jξponto T0(p) is non–zero. Then T0(p) is a real subspace of TpCHnby Corollary 7.5. Since dim T0(p) = 2n−3, this is impossible for n > 3 and we must have n= 3. As ξp∈T0(p)⊥it follows that Jξp∈T0(p). Since orthogonal projection onto subbundles is a continuous map, this must hold on an open neighborhood Uof pin M. Therefore, Uis a Hopf hypersurface in CH3with three distinct constant principal curvatures 0, −1/2 and 1/2. According to Theorem 6.5 such a hypersurface does not exist. We conclude that the orthogonal projection of the Hopf vector field Jξ onto T0is zero everywhere. Now define M+as the set of all points p∈Mat which the orthogonal projections of Jξponto T−1/2(p) and T1/2(p) are both non–zero. Clearly, M+is an open subset of M. Using again the classification of Theorem 6.5 we see that M+is non–empty. Let Xand Ybe local unit vector fields on Mwith X∈Γ(T−1/2) and Y∈Γ(T1/2). Then we can write Jξ =aX +bY with a, b ∈Rsuch that a2+b2= 1. We may assume that Xand Yare chosen such that a, b ≥0. As we have seen above, T0(p) cannot be a real subspace at any point p∈M. Thus there exists a non–zero vector field U∈Γ(T0) such that JU ∈Γ(T0). Since ¯ ∇J= 0 we have ¯ ∇UJξ =J¯ ∇Uξ=JSU = 0, and thus Lemma 7.3 implies 0 = UhJU, Jξi=h∇UJU, Jξi=ah∇UJU, Xi+bh∇UJU, Y i=1 2(a2−b2)hU, Ui. This gives a2=b2and hence a=b= 1/√2. This shows that M+is a closed subset of M. As M+is open and non–empty, we see that M+=M. In particular, the length of the orthogonal projections of the Hopf vector field Jξ onto T−1/2and T1/2is constant and 140 6 Cohomogeneity one actions on the complex hyperbolic space equal to 1/√2. We now define Z=a(X−Y). Then the second fundamental form of M has the form of that in Theorem 6.16. Indeed, II(Z, Jξ) = −h¯ ∇ZJξ, ξiξ=hSZ, Jξiξ=−a2 2hX+Y, X +Yiξ=−1 2ξ , II(Jξ, Jξ) = −h¯ ∇JξJξ, ξiξ=hSJξ, Jξiξ=a2 2h−X+Y, X +Yiξ= 0, II(Z, Z) = −h¯ ∇ZZ, ξiξ=hSZ, Ziξ=a2 2hX+Y, X −Yiξ= 0, and II(U, X) = −h¯ ∇UX, ξi=hX, SUi= 0 for any U∈Γ(T0) and X∈Γ(TM). The result now follows from that theorem. 6.3.3 Cohomogeneity one actions with one non–totally geodesic singular orbit Let Hbe a closed subgroup of AN and consider the closed subgroup N0 K(H)H⊂KAN, where N0 K(H) is the identity component of the normalizer NK(H) = {k∈K:kHk−1⊂H} of Hin K. Let F=H·obe the orbit of Hthrough o. Then, F= (N0 K(H)H)·oand the following result holds [12]. Theorem 6.9. Let hbe the Lie algebra of H. Assume hcan be written in the form h=a⊕z⊕w, where w⊥is a linear vector subspace of vof dimension ≥2and constant K¨ahler angle ϕ. Then N0 K(H)Hacts on CHnwith cohomogeneity one and Fis a singular orbit of that action. Furthermore, if ϕ∈(0, π/2] then Fis not totally geodesic in CHn. Let V⊂Cnbe a linear subspace. Let v∈Vbe a non–zero vector. The K¨ahler angle of Vwith respect to vis the angle ϕ(v)∈[0, π/2] between Vand the real span of iv. Thus, ϕ(v)∈[0, π/2] is determined by requiring that (cos ϕ(v))kvkis the length of the orthogonal projection of iv onto V. We say that Vhas constant K¨ahler angle ϕif ϕ(v) = ϕfor all non–zero vectors v∈V. Linear subspaces with constant K¨ahler angle are of interest in what follows so we first derive some results that will be used later. Let V⊂Cna linear subspace with constant K¨ahler angle ϕ∈[0, π/2]. We denote by J the endomorphism of Cnconsisting of multiplying by the imaginary unit, that is, Jv =iv for all v∈Cn. If ϕ= 0 then Vis said to be a complex subspace of Cn. This is equivalent to JV ⊂V. If ϕ=π/2 then Vis a real subspace of Cn. In this case JV ⊂CnªV. Let CVbe the minimal complex vector subspace of Cncontaining Vand let V⊥= CVªV. We denote by π:CV→Vand σ:CV→V⊥the orthogonal projections onto Vand V⊥, respectively. We define P=πJ and F=σJ. If ϕ= 0 we have CV=V, π= IdCV,σ= 0, P=Jand F= 0. If ϕ=π/2 we have the orthogonal direct sum decomposition CV=V⊕JV and P=J σ,F=J π. In what follows we study the non–trivial case ϕ∈(0, π/2). 6.3.3 Cohomogeneity one actions with one non–totally geodesic singular orbit 141 Lemma 6.10. Let V⊂Cnbe a linear subspace with constant K¨ahler angle ϕ∈(0, π/2). Then P2=−(cos2ϕ)π−PFσ, F2=−FPπ−(cos2ϕ)σ, PF =−(sin2ϕ)π+PFσ, FP =FPπ −(sin2ϕ)σ. Proof. Let x∈V. Since Vhas constant K¨ahler angle ϕ, by definition, hPx, Pxi= (cos2ϕ)hx, xi. Iterating the equality hP2x, P2xi= (cos2ϕ)hPx, Pxi= (cos4ϕ)hx, xi. On the other hand, hP2x, xi=hJPx, xi=−hPx, Jxi=−hPx, Pxi=−(cos2ϕ)hx, xi. Since P2xhas the same length as the component of P2xin the direction of xwe obtain P2x= −(cos2ϕ)x. Then, P2π=−(cos2ϕ)π. Now we have −x=J2x=P2x+FPx +PFx +F2x. Taking the component in V⊥we get F2x=−FPx, that is, F2π=−FPπ. Taking the component in Vwe get −x=P2x+PFx =−(cos2ϕ)x+PFx. Thus, PFx =−(sin2ϕ)x, which implies PFπ = −(sin2ϕ)π. Using the above relations we have JFPx =PFPx+F2Px =−(sin2ϕ)Px+(cos2ϕ)Fx. Also, hFx, FPxi=−hx, JFPxi=−hx, PFPxi= (sin2ϕ)hx, Pxi= 0. Altogether this means that for any non–zero vector x∈Vthe vectors x,Px,Fx and FPx are orthogonal and span a complex vector subspace of CV. Moreover, x, Px ∈Vand Fx, FPx ∈V⊥. A similar argument in CVª(Rx⊕RPx ⊕RFx ⊕RFPx) shows that there exist non– zero vectors x1, . . . , xk∈Vsuch that {x1, Px1, . . . , xk, Pxk}is an orthogonal basis of V and {Fx1, FPx1, . . . , Fxk, FPxk}is an orthogonal basis of V⊥. In particular this implies that the dimension of Vis even. Now let y∈V⊥. We observe that Fy is the projection of Jy onto V⊥. The existence of the previous basis of CVshows that we can write y=aFx +bFPx for some x∈Vand a, b ∈R. For any z∈Vwe have hFz, Fzi=hJz, Jzi−hPz, Pzi= (1 −cos2ϕ)hz, zi= (sin2ϕ)hz, zi. This, the above results and the fact that hFPx, Fxi= 0 implies hFy, Fyi=a2hF2x, F2xi+ 2abhF2x, F2Pxi+b2hF2Px, F2Pxi =a2(sin2ϕ)(cos2ϕ)hx, xi+ 2ab(cos2ϕ)hFPx, Fxi+b2(cos4ϕ)(sin2ϕ)hx, xi = (cos2ϕ)³a2hFx, Fxi+b2hFPx,FPxi´= (cos2ϕ)hy, yi, which shows that V⊥has constant K¨ahler angle. Reversing the roles of Pand Fwe get F2σ=−(cos2ϕ)σ,FPσ =−(sin2ϕ)σ,P2σ=−FPσ. Altogether this gives the result. The proof of the previous lemma implies Corollary 6.11. Let V⊂Cnbe a vector subspace with constant K¨ahler angle ϕ∈(0, π/2). Then Vhas even dimension, let us say 2kand there exist non–zero vectors x1, . . . , xk∈V such that {x1, Px1, . . . , xk, Pxk}is an orthogonal basis of V. Moreover, CVªVhas also constant K¨ahler angle ϕ. An easy consequence of the definition allows us to calculate the inner product of the orthogonal projections of a vector onto Vand V⊥. 142 6 Cohomogeneity one actions on the complex hyperbolic space Corollary 6.12. Let V⊂Cna linear subspace with constant K¨ahler angle ϕ∈[0, π/2]. We have (i) If x, y ∈Vthen hPx, Pyi= (cos2ϕ)hx, yiand hFx, Fyi= (sin2ϕ)hx, yi. (ii) If x, y ∈V⊥then hPx, Pyi= (sin2ϕ)hx, yiand hFx, Fyi= (cos2ϕ)hx, yi. Proof. For ϕ= 0 and ϕ=π/2 the result follows immediately. Let ϕ∈(0, π/2). Since Vhas constant K¨ahler angle ϕ, for any x∈Vwe have hPx, Pxi= (cos2ϕ)hx, xi. Polarization of this equality implies hPx, Pyi= (cos2ϕ)hx, yifor all x, y ∈V. Hence hFx, Fyi= hJx, Jyi−hPx, Pyi= (1 −cos2ϕ)hx, yi= (sin2ϕ)hx, yi. This proves (i). Statement (ii) follows easily after taking into account that V⊥has also constant K¨ahler angle ϕand the roles of Pand Fare reversed. We give a geometric construction of the singular orbits of cohomogeneity one actions in CHnwith one non–totally geodesic singular orbit [15]. Let KAN be the Iwasawa decomposition with respect to o∈CHnand x∈CHn(∞), and let k⊕a⊕nbe the corresponding decomposition on Lie algebra level. The nilpotent algebra nis decomposed into n=z⊕vas described in Section 6.2. Let wbe a linear subspace of vsuch that w⊥=vªwhas constant K¨ahler angle ϕ∈[0, π/2]. Then h=a⊕z⊕wis a subalgebra of a⊕z⊕vof codimension k. Denote by Hthe closed subgroup of AN with Lie algebra hand by N0 K(H) the identity component of the normalizer of Hin K. Then G=N0 K(H)H⊂KAN acts on CHnwith cohomogeneity one. We denote by W2n−k ϕthe orbit of Gthrough o. For all g∈N0 K(H) we have g(H·o) = g(H·g−1o) = (gHg−1)·o⊂H·o, and hence W2n−k ϕ=H·o. If k= 1, then obviously ϕ=π/2, and the orbits of this action form a Riemannian foliation on CHnwhich is the solvable foliation described in the previous subsection. In this case the ruled minimal orbit of this foliation W2n−1is exactly W2n−1 π/2. In general, if ϕ=π/2 we denote W2n−k=W2n−k π/2. If k > 1, then W2n−k ϕhas codimension k, and all other orbits are the tubes around it. If ϕ= 0, then kis even, say k= 2j, and W2n−k 0is a totally geodesic CHn−j⊂CHn. In this case the action of Gis orbit equivalent to the action in Theorem 6.4 (i). For this reason we assume ϕ > 0 from now on. Any two Iwasawa decompositions of KAN are conjugate, and any two linear subspaces of vwith the same dimension and the same K¨ahler angle are conjugate by g∗= Ad(g) for some gin the normalizer of Ain K[6]. As a consequence any two submanifolds W2n−k ϕ and W2n−j φwith k=jand ϕ=φare holomorphically congruent. We now study the geometry of W2n−k ϕin more detail. The maximal complex subspace cof his a subalgebra and the closed subgroup Hcof Hwith Lie algebra cacts on W2n−k ϕ isometrically by left translations. The orbit Hc·ois a totally geodesic CHn−k⊂CHn. Note that the complex dimension of cis n−k. By identifying W2n−k ϕwith Hequipped with the induced left–invariant Riemannian metric, it follows now that W2n−k ϕis ruled by totally geodesic CHn−k⊂CHn. The Lie algebra a⊕ncan be decomposed orthogonally into a⊕n=c⊕d⊕w⊥. 6.3.3 Cohomogeneity one actions with one non–totally geodesic singular orbit 143 We denote by C,Dand W⊥the corresponding left–invariant distributions on CHnalong W2n−k ϕ. As we have seen above, Cis autoparallel and the integral submanifolds are totally geodesic CHn−k⊂CHn. The distribution W⊥is just the normal bundle T⊥W2n−k ϕ. We give a geometric description of the submanifold W2n−k ϕ. Proposition 6.13. The submanifold W2n−k ϕhas the following properties: (i) The maximal holomorphic subbundle Cof TW2n−k ϕis integrable and the leaves of the induced foliation on W2n−k ϕare totally geodesic CHn−k⊂CHn. (ii) The following statements are equivalent: (a) The distribution Don W2n−k ϕis integrable. (b) The distribution RA⊕Don W2n−k ϕis integrable. (c) ϕ=π/2. In this case the leaves of the foliation on W2n−kinduced by RA⊕Dare totally geodesic RHk+1 ⊂CHnand the leaves of the foliation on W2n−kinduced by Dare horospheres with center xin these totally geodesic RHk+1 ⊂CHn. (iii) The left-invariant subbundle W⊥of TCHnalong W2n−k ϕis the normal bundle of W2n−k ϕ. (iv) For each non–zero ξ∈w⊥the left–invariant distribution RA⊕RPξ along W2n−k ϕ is integrable and the leaves of the induced foliation on W2n−k ϕare totally geodesic RH2⊂CHn. (v) For each non–zero ξ∈w⊥the left-invariant distribution RPξ on W2n−k ϕis integrable and the leaves of the induced foliation on W2n−k ϕare horocycles with center xin the totally geodesic RH2⊂CHngiven by the distribution RA⊕RPξ. Proof. We use the formulas and notations as described in Section 6.2. Statement (i) follows immediately from the expression of the Levi–Civita connection of CHnfor left–invariant vector fields and the fact that the only complex totally geodesic submanifolds of CHnare complex hyperbolic spaces. Using the formulas for the Lie bracket of left–invariant vector fields in CHnwe get [aA +U, bA +V] = a 2V−b 2U+ [U, V ] and [U, V ] = hJU, V iZ. for all aA +U, bA +V∈RA⊕D. This shows that RA⊕Dis integrable if and only if Dis integrable if and only if Dis real, that is, ϕ=π/2. In this case the Levi–Civita connection yields ¯ ∇aA+U(bA +V) = 1 2hU, V iA−b 2U∈RA⊕D 144 6 Cohomogeneity one actions on the complex hyperbolic space for all aA +U, bA +V∈RA⊕D. This shows that RA⊕Dis autoparallel and its leaves are totally geodesic real submanifolds of CHn. The only real totally geodesic submanifolds of CHnare the real hyperbolic spaces. For all U, V ∈Dwe have ¯ ∇UV=1 2hU, V iAand ¯ ∇UA=−1 2U, which implies that the leaves of Dare spherical hypersurfaces of the corresponding real hyperbolic spaces (Theorem 6.6). Since the sectional curvature of a totally geodesic real hyperbolic subspace is −1/4, and the mean curvature vector of any leaf of Dis (1/2)A, it follows that the leaves of Dare horospheres centered at xin the real hyperbolic subspaces. This finishes the proof of (ii). Statement (iii) holds by construction. For any aA +xPξ, bA +yPξ ∈RA⊕RPξ we have ¯ ∇aA+xPξ(bA +yPξ) = xy 2(sin2ϕ)A−bx 2Pξ ∈RA⊕RPξ. From this, we easily get the assertion (iv) using Theorem 6.6. Finally, define Uξ=Pξ/ sin(ϕ). Then the expression of the Levi–Civita connection for left–invariant metrics implies ¯ ∇UξUξ=1 2Aand ¯ ∇Uξ¯ ∇UξUξ=−1 4Uξ. Since the real hyperbolic planes in (iv) have constant sectional curvature −1/4, this shows that the integral curves of Uξare horocycles with center xin the corresponding real hyperbolic planes. This proves (v). The above properties are characteristic of W2n−k ϕ. Any other submanifold of CHnwith these properties is holomorphically congruent to some W2n−k ϕas the following result shows. Afterwards we will see that, in fact, all the information of W2n−k ϕis encoded in its second fundamental form. Corollary 6.14. Let k∈ {1, . . . , n −1}, and fix a totally geodesic CHn−k⊂CHnand points o∈CHn−kand x∈CHn−k(∞). Let KAN be the Iwasawa decomposition of SU(1, n)with respect to oand x, and let H0be the subgroup of AN which acts simply transitively on CHn−k. Next, let Vbe a subspace of T⊥ oCHn−kwith constant K¨ahler angle ϕ∈(0, π/2] such that CV=T⊥ oCHn−k. Left translation of Vby H0to all points in CHn−k determines a subbundle Vof the normal bundle T⊥CHn−k. At each point p∈CHn−k attach the horocycles determined by xand the linear lines in Vp. The resulting subset M of CHnis holomorphically congruent to the ruled submanifold W2n−k ϕ. Proof. Let W2n−k ϕbe the ruled minimal submanifold of CHnconstructed from the Iwasawa decomposition KAN associated with xand oand the choice of w⊥=T⊥ oCHn−kªV. We use the above notations. From Proposition 6.13 we already know that M⊂W2n−k ϕ. It suffices to prove that W2n−k ϕ⊂M. 6.3.3 Cohomogeneity one actions with one non–totally geodesic singular orbit 145 Let p∈W2n−k ϕ. There exists an isometry s∈Hwith p=s(o). Then there is a unique vector Xin the Lie algebra hof Hsuch that s= Exps(X). We can write X=aA +zZ +U+Vwith some U∈c,V∈dand a, z ∈R. Note that [V, U] = 0 because they are complex orthogonal. We now define g= Exps³ρ³a 2´V´and h= Exps(aA +zZ +U). Note that h∈H0. Using the description of CHngiven in Section 6.2 we get gh = Exps³ρ³a 2´V´Exps(aA +zZ +U) =³0,Expn³ρ³a 2´V´´·³a, Expn³ρ(a)z Z +ρ³a 2´U´´ =µa, Expnµρ(a)z Z +ρ³a 2´U+ρ³a 2´V+1 2[V, U]¶¶ = Exps(aA +zZ +U+V) = s. By construction, h(o)∈CHn−kand s(o) = g(h(o)) is on the horocyle with center x through h(o) tangent to RV. From this we conclude that W2n−k ϕ⊂M. Altogether this implies M=W2n−k ϕand the result follows. Next, we calculate the second fundamental form of W2n−k ϕ. Proposition 6.15. The second fundamental form of W2n−k ϕis given by the formula II¡aA +xZ +U+Pξ, bA +yZ +V+Pη¢=−sin2ϕ 2¡yξ +xη¢ for any U, V ∈TW2n−k ϕª(RA⊕RZ),ξ, η ∈T⊥W2n−k ϕand a, b, x, y ∈R. Thus II is given by the trivial bilinear extension of 2II(Z, Pξ) = −(sin2ϕ)ξfor any ξ∈T⊥W2n−k ϕ. Proof. Since A,Z,U,V,Pξ and Pη are tangent to W2n−k ϕ, the normal component of the Levi–Civita connection reduces to II¡aA +xZ +U+Pξ, bA +yZ +V+Pη¢=−¡¯ ∇aA+xZ+U+Pξ(bA +yZ +V+Pη)¢⊥ =³y 2JPξ +x 2JPη´⊥=y 2FPξ +x 2FPη. Since FP|T⊥W2n−k ϕ=−(sin2ϕ) IdT⊥W2n−k ϕby Lemma 6.10, the result follows. The second fundamental form of W2n−k ϕand the fact that its normal bundle has constant K¨ahler angle ϕare enough to characterize W2n−k ϕamong all the submanifolds of CHn.