Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms

NON-HOPF REAL HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES IN COMPLEX SPACE FORMS JOS´ E CARLOS D´ IAZ-RAMOS AND MIGUEL DOM´ INGUEZ-V´ AZQUEZ Abstract. We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Br¨uck. In particular, they are open parts of homogenous ones. 1. Introduction A homogeneous submanifold of a Riemannian manifold is an orbit of the action of a closed subgroup of the isometry group of the ambient manifold. One of the aims of submanifold geometry is to classify homogeneous submanifolds of a given manifold and to characterize them in terms of geometric data. Of particular interest are homogeneous hypersurfaces, which arise as principal orbits of cohomogeneity one actions. Obviously, homogeneous hypersurfaces have constant principal curvatures, that is, the eigenvalues of their shape operator are constant. It is an outstanding problem to determine under which conditions hypersurfaces with constant principal curvatures are open parts of homogeneous ones. In spaces of constant curvature, a hypersurface has constant principal curvatures if and only if it is isoparametric. The classification of isoparametric hypersurfaces was achieved by Segre [20] in Euclidean spaces and by Cartan [9] in real hyperbolic spaces. They all are open parts of homogeneous ones. The situation is more involved in spheres. Cartan classified hypersurfaces with g∈ {1,2,3}constant principal curvatures in spheres. Subsequently, Hsiang and Lawson [12] classified homogeneous hypersurfaces in spheres; they have g∈ {1,2,3,4,6}principal curvatures. Then, M¨unzner [18] showed that g∈ {1,2,3,4,6}for isoparametric hypersurfaces in general. Surprisingly, for g= 4 there are isoparametric hypersurfaces that are not homogeneous [13]. Recently, Cecil, Chi and Jensen [10], and Immervoll [14] showed that, with a few possible exceptions, hypersurfaces with g= 4 constant principal curvatures are among the known homogeneous and inhomogeneous examples. Some progress has been made for g= 6 by Abresch [1] and Dorfmeister and Neher [11], but the problem remains open in full generality. See [24] for a survey. 1991 Mathematics Subject Classification. Primary 53C40, Secondary 53C55, 53C35. Key words and phrases. Hopf hypersurfaces, homogeneous hypersurfaces, constant principal curvatures. The first author has been supported by a Marie-Curie European Reintegration Grant (PERG04-GA-2008-239162). The second author has been supported by the FPU programme of the Spanish Government. Both authors have been supported by project MTM2009-07756 (Spain). 1 2 J.C. D´ IAZ-RAMOS AND M. DOM´ INGUEZ-V´ AZQUEZ The problem is even more difficult in complex space forms. See [19] for a survey on this and related topics. By c6= 0 we denote the constant holomorphic sectional curvature of a complex space form; thus, if c > 0 (resp. c < 0) we have a complex projective (resp. hyperbolic) space CPn(c) (resp. CHn(c)). We denote by Jits K¨ahler structure. Let Mbe a real hypersurface of a complex space form and ξ a (local) unit normal vector field. Then, Jξ is tangent to Mand is called the Hopf vector field of M. The hypersurface Mis said to be Hopf if Jξ is a principal curvature vector field. The motivation for our work is to address the classification of real hypersurfaces with constant principal curvatures in complex space forms. We briefly summarize the current state of the problem. Assume Mis a real hypersurface of a complex space form with gdistinct constant principal curvatures. For p∈Mdenote by h(p) the number of nontrivial projections of Jξponto the principal curvature spaces of M. Clearly, this function is integer-valued and Mis Hopf if and only if h= 1. The classification of homogeneous real hypersurfaces in complex projective spaces CPn(c) was derived by Takagi [21]. It follows from this classification that g∈ {2,3,5}. A remarkable feature of homogeneous real hypersurfaces in CPn(c) is that they are Hopf. Subsequently, Takagi classified real hypersurfaces with g∈ {2,3}constant principal curvatures [22], [23] ([25] for n= 2, g= 3). It follows from his work that they all are Hopf and open parts of homogeneous ones. Kimura [15] classified Hopf real hypersurfaces with constant principal curvatures in CPnand showed that these are open parts of homogeneous ones. No examples are known of real hypersurfaces with constant principal curvatures in CPn(c) with h > 1. Surprisingly, in CHn(c) there are non-Hopf homogeneous real hypersurfaces. The first example was discovered by Lohnherr [16] and further examples were given by Berndt and Br¨uck [3], [4]. We refer to §2.2 for a brief introduction and to [7] for a deeper study of their geometry. Berndt and Tamaru obtained in [8] the classification of cohomogeneity one actions on CHn(c). The number of principal curvatures of the homogeneous examples is g∈ {2,3,4,5}. Montiel [17] classified real hypersurfaces with g= 2 constant principal curvatures in CHn(c) (n≥3). Berndt and the first author solved the case g= 3 and g= 2, n= 2 [5], [6]. It follows from [17] that h= 1 when g= 2, and from [5] and [6] we get h≤2 if g= 3. Hopf real hypersurfaces with constant principal curvatures in CHn(c) were classified by Berndt [2] and they all are open parts of homogeneous ones. To our knowledge, [5] and [6] are the first classifications of this kind involving non-Hopf real hypersurfaces. Nothing is known about hif g≥4. Our aim in this paper is to carry out the next natural step after Berndt and Kimura’s classification of Hopf real hypersurfaces with constant principal curvatures in CPn(c) and CHn(c). Thus, we classify real hypersurfaces with constant principal curvatures whose Hopf vector field Jξ has h= 2 nontrivial projections onto the principal curvature spaces. Main Theorem. We have: (a) There are no real hypersurfaces with constant principal curvatures in CPn(c), n≥2, whose Hopf vector field has h= 2 nontrivial projections onto the principal curvature spaces. (b) Let Mbe a connected real hypersurface in CHn(c),n≥2, with constant principal curvatures and whose Hopf vector field has h= 2 nontrivial projections onto the principal curvature spaces of M. Then, Mhas g∈ {3,4}principal curvatures and is holomorphically congruent to an open part of: NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 3 (i) a ruled minimal real hypersurface W2n−1⊂CHn(c)or one of the equidistant hypersurfaces to W2n−1, or (ii) a tube around a ruled minimal Berndt-Br¨uck submanifold with totally real normal bundle W2n−k⊂CHn(c), for some k∈ {2, . . . , n −1}. In particular, Mis an open part of a homogeneous real hypersurface of CHn(c). The ruled minimal submanifolds W2n−k⊂CHn(c) are homogeneous and have totally real normal bundle of rank k∈ {1, . . . , n −1}. Actually, W2n−1was discovered by Lohnherr [16]. Then, Berndt studied the geometry of the equidistant hypersurfaces to W2n−1[3]. This construction was generalized by Berndt and Br¨uck in [4]. Both W2n−1and any of its equidistant hypersurfaces have g= 3 principal curvatures. The tubes around W2n−k,k∈ {2, . . . , n −1}have g= 4 principal curvatures if r6= (1/√−c) log(2 + √3) and g= 3 principal curvatures if r= (1/√−c) log(2 + √3). See [7] for a detailed description. The proof is as follows. First we use the Gauss and Codazzi equations to derive some algebraic properties of the eigenvalue structure of the shape operator. The methods used for this are similar to those of [5], although a bit more general. We would like to emphasize that whenever we use a method similar to one in [5] we explicitly point it out and skip the details as much as possible. On the other hand, we focus on the new techniques and results, especially on Subsection 3.4. The most crucial step of the proof is to show that the number gof constant principal curvatures satisfies g≤4. For this we use a novel approach based on the study of some inequalities satisfied by the principal curvatures. Using standard Jacobi field theory one can deduce the geometry of the focal submanifolds of these hypersurfaces and then the result follows from a rigidity result in [7]. The paper is organized as follows. In Section 2 we introduce the basic elements of our paper. Subsection 2.1 is devoted to present the equations of submanifold geometry that we will use in the rest of the paper. In §2.2 we briefly describe the ruled minimal Berndt-Br¨uck submanifolds W2n−k. We prove our Main Theorem in Section 3. The proof is divided in several steps. Some vector fields and functions arise naturally in our proof (§3.1 and §3.2). We get some of their properties in Subsection 3.3. In §3.4 we show that the number gof principal curvatures satisfies g∈ {3,4}. We summarize all the eigenvalue structure in §3.5. In Subsection 3.6 we use standard Jacobi field theory to finish the proof of the Main Theorem. 2. Preliminaries In this section we introduce the basic notation of this paper. We write down the Gauss and Codazzi equations of a hypersurface in a complex space form and derive some basic consequences. Then, we briefly mention how the examples of the Main Theorem are constructed. 2.1. The equations of a hypersurface. Let ¯ M(c) be a complex space form of constant holomorphic sectional curvature c6= 0 and complex dimension n. If c > 0 then ¯ M(c) is a complex projective space CPn(c) of constant holomorphic sectional curvature c. Analogously, if c < 0 then ¯ M(c) is a complex hyperbolic space CHn(c). We denote by h·,·i its inner product, by Jits K¨ahler structure, and by ¯ ∇its Levi-Civita connection. The curvature tensor is defined by ¯ R(X, Y ) = 4 J.C. D´ IAZ-RAMOS AND M. DOM´ INGUEZ-V´ AZQUEZ [¯ ∇X,¯ ∇Y]−¯ ∇[X,Y ], so in this case we have ¯ R(X, Y )Z=c 4(hY, ZiX−hX, ZiY+hJY, ZiJX −hJX, ZiJY −2hJX, Y iJZ). Let Mbe a connected submanifold of ¯ M(c). We denote by ∇and Rits Levi- Civita connection and its curvature tensor respectively. By TM and νM we denote the tangent and normal bundles of M. We use the symbol Γ(·) to refer to the smooth sections of any vector bundle. Let X, Y, Z, W ∈Γ(TM) and ξ∈Γ(νM). The second fundamental form II of Mis defined by the Gauss formula as ¯ ∇XY= ∇XY+II(X, Y ). The Weingarten formula is then written as ¯ ∇Xξ=−SξX+∇⊥ Xξ, where Sξis the shape operator with respect to ξand ∇⊥is the induced normal connection on νM. The second fundamental form and the shape operator are related by hSξX, Y i=hII(X, Y ), ξi. Now let Mbe a connected real hypersurface of ¯ M(c). The word ‘real’ emphasizes the fact that the real codimension is one. Fix ξ∈Γ(νM) a (local) unit normal vector field. We write Sinstead of Sξ. The Gauss formula can be rewritten as ¯ ∇XY=∇XY+hSX, Y iξ, and hence, the Weingarten formula is SX =−¯ ∇Xξ. Moreover, the Gauss and Codazzi equations for a hypersurface are h¯ R(X, Y )Z, W i=hR(X, Y )Z, Wi−hSY, ZihSX, Wi+hSX, ZihSY, Wi,and h¯ R(X, Y )Z, ξi=h(∇XS)Y−(∇YS)X, Zi. We assume from now on that Mhas constant principal curvatures, that is, the eigenvalues of the shape operator Sare constant. For each principal curvature λof Mwe denote by Tλthe distribution on Mformed by the principal curvature spaces of λalong M. The Codazzi equation implies (see [5, Section 2] for a proof) Lemma 2.1. (i) Let p∈M. If the orthogonal projection of Jξponto Tα(p)is nonzero, then Tα(p)is a real subspace of Tp¯ M(c), that is, JTα(p)is orthogonal to Tα(p). (ii) Let X, Y ∈Γ(Tα)and Z∈Γ(Tβ)with α6=β. Then h∇XY, Zi=c 4(α−β)(hJY, ZihX, Jξi+hJX, Y ihZ, Jξi+ 2hJX, ZihY, Jξi). (iii) Let X∈Γ(Tα),Y∈Γ(Tβ)and Z∈Γ(Tγ). Then h¯ R(X, Y )Z, ξi= (β−γ)h∇XY, Zi−(α−γ)h∇YX, Zi. The Gauss equation implies (again, see [5, Lemma 4] for a proof) Lemma 2.2. Let X∈Γ(Tα)and Y∈Γ(Tβ), with α6=β, be unit vector fields. Then 0=(β−α)(−c−4αβ −2chJX, Y i2+ 8h∇XY, ∇YXi−4h∇XX, ∇YYi) −4chJX, Y i(XhY, Jξi+YhX, Jξi) −chX, Jξi(3YhJX, Y i+h∇YX, JY i−2h∇XY, JY i) −chY, Jξi(3XhJX, Y i−h∇XY, JXi+ 2h∇YX, JXi). NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 5 2.2. Discussion of examples. Part (a) of the Main Theorem states that there are no examples of real hypersurfaces with constant principal curvatures in CPn(c) whose Hopf vector field has h= 2 nontrivial projections onto the principal curvature spaces of M. Thus, we will focus on describing briefly the examples of part (b) of the Main Theorem. These examples where first constructed in [4] and their geometry was studied in [7]. The connected simple Lie group G=SU(1, n) acts transitively on CHn(c). Fix a point o∈CHn(c) and let Kbe the isotropy group of Gat o. The subgroup Kof Gis isomorphic to S(U(1)U(n)). Furthermore, (G, K) is a symmetric pair and CHn(c) may be identified with the quotient G/K. Write gfor the Lie algebra of Gand kfor the Lie algebra of K. Let g=k⊕pbe the Cartan decomposition of gwith respect to o∈CHn(c). We choose a maximal abelian subspace aof p; then, dim a= 1 since CHn(c) has rank one. Let g=g−2α⊕g−α⊕g0⊕gα⊕g2αbe the restricted root space decomposition of gwith respect to aand assume that αis a positive root. Then, n= gα⊕g2αis a 2-step nilpotent subalgebra of gisomorphic to the (2n−1)-dimensional Heisenberg algebra. Furthermore, g=k⊕a⊕nis an Iwasawa decomposition of g. If Aand Ndenote the connected subgroups of Gwhose Lie algebras are aand n, then G=KAN is an Iwasawa decomposition of G. The solvable group AN is simply connected and acts simply transitively on CHn(c). Thus, we can identify a⊕n with ToCHn(c). The Riemannian metric of CHn(c) induces a left-invariant metric on AN which makes AN isometric to CHn(c). Similarly, the complex structure Jon ToCHn(c) induces a complex structure on a⊕nwhich we also denote by J. We have Ja=g2α, and gαis J-invariant. Let B∈abe a unit vector and define Z=JB ∈g2α. Let wbe a linear subspace of gαsuch that the orthogonal complement w⊥= gαªwof win gαhas constant K¨ahler angle ϕ∈(0, π/2], that is, the angle between Jv and w⊥is ϕfor all nonzero v∈w⊥. Then, ϕ=π/2 if and only if w⊥is real, or equivalently, if and only if Jw⊥is orthogonal to w⊥. Let kbe the dimension of w⊥. Then, s=a⊕w⊕g2αis a subalgebra of a⊕n. Let Sbe the connected simply connected subgroup of AN whose Lie algebra is s. We define the Berndt-Br¨uck submanifolds as [4] (see [16] for k= 1) W2n−k ϕ=S·o, and W2n−k=W2n−k π/2. The Berndt-Br¨uck submanifolds W2n−k ϕare homogeneous, have normal bundle of rank kand constant K¨ahler angle ϕ∈(0, π/2], and their second fundamental form II is given by the trivial symmetric bilinear extension of II(Z, P ξ) = (sin2(ϕ)√−c/2)ξfor all ξ∈w⊥, where Pξ is the orthogonal projection of Jξ onto TW2n−k ϕ. In particular, the submanifolds W2n−k ϕare minimal, and ruled by the totally geodesic complex hyperbolic subspaces determined by their maximal holomorphic distribution. If ϕ=π/2 then P=Jand the Berndt-Br¨uck submanifolds have totally real normal bundle. Conversely [7, Theorem 1], Theorem 2.3. Let Mbe a (2n−k)-dimensional connected submanifold in CHn(c), n≥2, with normal bundle νM of constant K¨ahler angle ϕ∈(0, π/2]. Assume that there exists a unit vector field Ztangent to the maximal holomorphic subbundle of TM such that the second fundamental form II of Mis given by the trivial symmetric bilinear extension of II(Z, Pξ) = sin2(ϕ)√−c 2ξ 6 J.C. D´ IAZ-RAMOS AND M. DOM´ INGUEZ-V´ AZQUEZ for all ξ∈Γ(νM). Then Mis holomorphically congruent to an open part of the ruled minimal submanifold W2n−k ϕ. In particular, the Berndt-Br¨uck submanifolds W2n−kare determined by the equation II(Z, Jξ) = (√−c/2)ξand the fact that their normal bundle is totally real. Geometrically, they are constructed in the following way. Fix a horosphere Hin a totally geodesic real hyperbolic space RHk+1(c)⊂CHn(c). Attach at each point the totally geodesic CHn−k(c) which is tangent to the orthogonal complement of the complex span of the tangent space of Hat p. The resulting submanifold is congruent to W2n−k. Let N0 K(S) denote the connected component of the identity transformation of the normalizer of Sin K. Then, N0 K(S)Sacts on CHn(c) with cohomogeneity one and W2n−k ϕ=N0 K(S)S·o. If k > 1, then the principal orbits of N0 K(S)Sare tubes around W2n−k ϕ. If k= 1, then ϕ=π/2, the action of N0 K(S)Sis orbit equivalent to the action of S, and its orbits form a homogeneous foliation on CHn(c) that was first studied in [3]. Let Mbe a principal orbit of N0 K(S)S. If ϕ∈(0, π/2) then the Hopf vector field of Mhas h= 3 nontrivial projections onto the principal curvature spaces of M. If ϕ=π/2, then the Hopf vector field of Mhas h= 2 nontrivial projections onto the principal curvature spaces of M. The objective of part (b) of the Main Theorem is to give a geometric characterization of the tubes around W2n−k,k∈ {2, . . . , n−1}, and the equidistant hypersurfaces to W2n−1. 3. Proof of the Main Theorem In this section we prove the Main Theorem. Our main goal is to describe accurately the eigenvalue structure of a real hypersurface in the conditions of the Main Theorem (Theorem 3.12). Then we finish the proof using standard Jacobi field theory (§3.6). 3.1. Notation and setup. Let Mbe a connected real hypersurface with g > 1 distinct constant principal curvatures in a complex space form ¯ M(c). Since the calculations that follow are local we may assume that we have a globally defined unit normal vector field ξ. We denote by λ1, . . . , λgthe principal curvatures of M. By assumption, the number of nontrivial projections of Jξ onto the principal curvature distributions Tλi,i∈ {1, . . . , g}, is h= 2. By relabeling the indices we may also assume that Jξ has nontrivial projection onto Tλ1and Tλ2. Hence, there exist unit vectors fields Ui∈Γ(Tλi), i∈ {1,2}, and positive smooth functions bi:M→R,i∈ {1,2}, such that Jξ =b1U1+b2U2. Obviously, b2 1+b2 2= 1. Moreover, Lemma 3.1. We have g≥3,hJU1, U2i= 0 and there exists a unit vector field A∈Γ(⊕g k=3Tλk)such that JUi= (−1)ibjA−biξ, (i, j ∈ {1,2}, i 6=j), JA =b2U1−b1U2. Proof. The proof is similar to that of [5, Lemma 7], so we just sketch it. We will assume in what follows i, j ∈ {1,2},i6=j, and k∈ {3, . . . , g}. NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 7 Since Tλi,i∈ {1,2}, is real by Lemma 2.1 (i) we can write JUi=hJUi, UjiUj+ Wij +Pg k=3 Wik −biξ, where Wij ∈Γ(TλjªRUj) and Wik ∈Γ(Tλk). (Here and henceforth, the symbol ªis used to denote orthogonal complement.) From Jξ =b1U1+b2U2we get −ξ=J2ξ=b2(hJU2, U1iU1+W21)+b1(hJU1, U2iU2+W12)+ g X k=3 (b1W1k+b2W2k)−ξ. Thus, g≥3, hJU1, U2i= 0, W12 =W21 = 0, and b1W1k+b2W2k= 0 for all k. If we define A∈Γ(⊕g k=3Tλk) by Pg k=3 Wik = (−1)ibjA, then the last equality implies Pg k=3 Wjk = (−1)jbiA(recall i, j ∈ {1,2},i6=j). This gives the desired expression for JUi,i∈ {1,2}. Finally, from b2 1+b2 2= 1 and −U1=J(JU1) = −b2JA−b1Jξ = −b2JA −U1+b2 2U1−b1b2U2we obtain JA =b2U1−b1U2.¤ 3.2. The vector field A.In view of Lemma 3.1 we may write A= g X k=3 Ak,with Ak∈Γ(Tλk), k ∈ {3, . . . g}. The aim of this subsection is to show that all but one Akare zero and hence we can assume for example that A∈Γ(Tλ3) (Proposition 3.3). The main difficulty here is the fact that gis not known. We start with the following Lemma 3.2. Let i, j ∈ {1,2}with i6=j. Then we have ∇UiUi= g X k=3 (−1)j3cb1b2 4(λk−λi)Ak,∇UiUj= g X k=3 (−1)jµλi−3cb2 i 4(λk−λi)¶Ak. Proof. Again, this is quite similar to [5, Lemma 8]. We assume i, j ∈ {1,2},i6=j, and k∈ {3, . . . , g}. Let Wi∈Γ(TλiªRUi) and Wk∈Γ(TλkªRAk). Since Uihas unit length, h∇UiUi, Uii= 0. Lemma 2.1 (ii) yields h∇UiUi, Uji= h∇UiUi, Wji=h∇UiUi, Wki= 0, and h∇UiUi, Aki= 3(−1)jcb1b2/(4(λk−λi)). From ¯ ∇J= 0, the Weingarten formula, and Lemma 3.1, we obtain hWi,¯ ∇UiJξi= −λihWi, JUii= 0. Hence, using Jξ =b1U1+b2U2, and Lemma 2.1 (ii), we get 0 = UihWi, Jξi=h∇UiWi, Jξi+hWi,¯ ∇UiJξi=−bih∇UiUi, Wii. Since bi6= 0 the expression for ∇UiUifollows. As Ujhas unit length, h∇UiUj, Uji= 0. From Lemma 2.1 (ii) we obtain h∇UiUj, Uii=h∇UiUj, Wii= 0. Now, the Weingarten formula and Lemma 3.1 imply hWj,¯ ∇UiJξi=−λihWj, JUii= 0, and thus, Lemma 2.1 (ii), yields 0 = UihWj, Jξi=h∇UiWj, Jξi+hWj,¯ ∇UiJξi=bjh∇UiWj, Uji. This implies h∇UiWj, Uji= 0. A similar calculation gives h∇UiWk, Uji= 0. Finally, by Lemma 2.1 (ii), and Lemma 3.1 we have 0 = UihAk, Jξi=h∇UiAk, Jξi+hAk,¯ ∇UiJξi = (−1)i3cb2 ibj 4(λk−λi)−bjh∇UiUj, Aki−(−1)iλibj, from where we get h∇UiUj, Aki. Altogether this yields the formula for ∇UiUj.¤ Now we can prove the main result of this section. Proposition 3.3. A∈Γ(Tλk)for some k∈ {3, . . . , g}. 8 J.C. D´ IAZ-RAMOS AND M. DOM´ INGUEZ-V´ AZQUEZ Proof. On the contrary, assume that there exists a point p∈Mand two distinct integers r, s ∈ {3, . . . , g}such that (Ar)p,(As)p6= 0. Hence, in a neighborhood of pwe have Ar, As6= 0 as well. We will work in that neighborhood from now on. Applying Lemma 2.1 (iii) to the vector fields U1,U2, and Ak,k∈ {r, s}, and using Lemma 3.2 we easily get (1) 3c(λ2−λk) 4(λ1−λk)b2 1+3c(λ1−λk) 4(λ2−λk)b2 2=−c 4−λ1(λ2−λk)−λ2(λ1−λk), k ∈ {r, s}. Together with b2 1+b2 2= 1, this yields a linear system of three equations with unknowns b2 1and b2 2. This system must be compatible. We show it is determined (that is, it has a unique solution). If it were not, the rank of the system would, at most, be one. In particular, ¯¯¯¯¯ 3c(λ2−λk) 4(λ1−λk) 3c(λ1−λk) 4(λ2−λk) 1 1 ¯¯¯¯¯ = 3c(λ2−λ1)(λ1+λ2−2λk) 4(λ1−λk)(λ2−λk)= 0, k ∈ {r, s}, which implies λ1+λ2−2λk= 0, k∈ {r, s}, and hence λr=λs, contradiction. We conclude that the above system is determined. Therefore, we can find an expression for b2 1and b2 2in terms of the principal curvatures and c. Since these are constant, it follows that b1and b2are constant. We take i, j ∈ {1,2},i6=j, and k∈ {r, s}. Since biis constant and Uihas unit length, using Jξ =b1U1+b2U2, the Weingarten formula, and Lemma 3.1 we get 0 = Ak(bi) = AkhUi, Jξi=h∇AkUi, Jξi+hUi,¯ ∇AkJξi=bjh∇AkUi, Uji−(−1)jbjλk, and thus, h∇AkUi, Uji= (−1)jλk. Taking this, Lemma 3.1, and Lemma 3.2 into account, Lemma 2.1 (iii) for Ak,U1and U2yields c 4(2b2 2−b2 1) = h¯ R(Ak, U1)U2, ξi= (λ1−λ2)λk+ (λk−λ2)µλ1−3cb2 1 4(λk−λ1)¶, for k∈ {r, s}. We can rearrange this as: (2) µc 4−3c(λk−λ2) 4(λk−λ1)¶b2 1−c 2b2 2= (λ2−λ1)λk+λ1(λ2−λk), k ∈ {r, s}. Hence, (1), (2), and b2 1+b2 2= 1 give a linear system of five equations with unknowns b2 1and b2 2. This system is compatible by assumption, so it has rank two. Then, all minors of order three of the augmented matrix of the system vanish. This implies (take (1), (2), and b2 1+b2 2= 1, with k∈ {r, s}, and then both equations in (2) and b2 1+b2 2= 1): 3c(λ1−λ2)2(−12λ2 k+ 8λ1λk+ 8λ2λk+c−4λ1λ2) 16(λ1−λk)(λk−λ2)= 0, k ∈ {r, s},(3) 3c(λ2−λ1)(λr−λs)(4λ2 1−4λrλ1−4λsλ1+c+ 2λ2λr+ 2λ2λs) 8(λ1−λr)(λ1−λs)= 0.(4) In particular, (3) implies −12λ2 k+ 8λ1λk+ 8λ2λk+c−4λ1λ2= 0. Putting k=r and k=s, and subtracting, we get 4(2λ1+ 2λ2−3λr−3λs)(λr−λs) = 0, from where we obtain λr+λs= 2(λ1+λ2)/3. Taking this into account, (4) gives (4λ2 1−4λ1λ2+ 4λ2 2+ 3c)/3 = 0. The discriminant of −12λ2 k+ 8λ1λk+ 8λ2λk+c− 4λ1λ2= 0 as a quadratic equation in λkis precisely 16(4λ2 1−4λ1λ2+ 4λ2 2+ 3c), so this discriminant vanishes. As a consequence, this quadratic equation has a unique NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 9 solution and hence λr=λs. This is a contradiction. Therefore, all but one Ak, k∈ {3, . . . , g}, are zero for each p. The result follows by continuity. ¤ 3.3. Some properties of the principal curvature spaces. In view of Proposition 3.3, we may assume from now on that A∈Γ(Tλ3). Moreover, we can choose an orientation on Mand a relabeling of the indices so that λ1< λ2,and λ3≥0. We will follow this convention from now on. First we calculate some covariant derivatives. Lemma 3.4. Let i, j ∈ {1,2}with i6=j. Then we have ∇UiUi= (−1)j3cb1b2 4(λ3−λi)A,(5) ∇UiUj= (−1)jµλi−3cb2 i 4(λ3−λi)¶A,(6) ∇UiA= (−1)i3cb1b2 4(λ3−λi)Ui+ (−1)iµλi−3cb2 i 4(λ3−λi)¶Uj,(7) ∇AUi=(−1)j λi−λjÃc(2b2 j−b2 i) 4+ (λj−λ3)µλi−3cb2 i 4(λ3−λi)¶!Uj,(8) ∇AA= 0.(9) Proof. The proof is similar to that of [5, Lemma 8]. Equations (5) and (6) are a direct consequence of Lemma 3.2 and Proposition 3.3. Assume i, j ∈ {1,2},i6=j, and k∈ {4, . . . , g}. Let Wi∈Γ(TλiªRUi), W3∈Γ(Tλ3ªRA) and Wk∈Γ(Tλk). According to (5) and (6), in order to prove (7) we have to show h∇UiA, Ai= 0 (obvious because Ais a unit vector field), and h∇UiA, Wli= 0 for all l∈ {1, . . . , g}. The latter follows from ¯ ∇J= 0, the Weingarten formula, Lemma 3.1, and (5), with 0 = UihJUi, Wli=h¯ ∇UiJUi, Wli+hJUi,¯ ∇UiWli =−h∇UiUi, JWli+ (−1)ibjhA, ∇UiWli−bihξ, ¯ ∇UiWli= (−1)jbjh∇UiA, Wli. We now prove (8). Obviously, h∇AUi, Uii= 0, and by Lemma 2.1 (ii) we get h∇AUi, Ai= 0. Applying Lemma 2.1 (iii) to A,Uiand Uj, using Lemma 3.1 and (6), gives c 4(−1)i(b2 i−2b2 j) = (λi−λj)h∇AUi, Uji−(λ3−λj)(−1)iµλi−3cb2 i 4(λ3−λi)¶, from where we get h∇AUi, Uji. For l∈ {j, 3, . . . , g}, a similar argument with Lemma 2.1 (iii) applied to A,Ui, and Wl, taking Lemma 3.1 and (7) into account, yields h∇AUi, Wli= 0. Finally, the previous equality (interchanging iand jand putting l=i) gives 0 = AhWi, Jξi=h∇AWi, Jξi+hWi,¯ ∇AJξi =bih∇AWi, Uii+bjh∇AWi, Uji−λ3hWi, JAi=−bih∇AUi, Wii. Altogether this proves (8). 16 J.C. D´ IAZ-RAMOS AND M. DOM´ INGUEZ-V´ AZQUEZ If we examine the proof of our theorem, so far we have actually shown that for any point p∈Mthere exists a neighborhood of pwhere the conclusion of Theorem 3.12 is satisfied. However, by the connectedness of Mand a continuity argument, it can be easily shown that Mis orientable and that the conclusion of Theorem 3.12 is satisfied globally. 3.6. Jacobi field theory and rigidity of focal submanifolds. In this last section we finish the proof of part (b) of the Main Theorem. Since we use standard Jacobi field theory, we provide the reader just with the fundamental details and skip the long calculations. According to [5] we just have to take care of the case g= 4. However, it is not much overload to deal with the two cases simultaneously, so for the sake of completeness we will do so in what follows. Let Mbe a real hypersurface of CHn(c) in the conditions of Theorem 3.12 (b). For r∈Rwe define the map Φr:M→CHn(c), p7→ expp(rξp), where exppis the Riemannian exponential map of CHn(c) at p. Then, Φr(M) is obtained by moving Ma distance ralong its normal direction. The singularities of Φrare the focal points of M. We will find a particular distance rfor which Φr ∗has constant rank, where Φr ∗denotes the differential of Φr. Then we will apply Theorem 2.3 to Φr(M) for this choice of r. This way, Φr(M) will be an open part of the ruled minimal Berndt-Br¨uck submanifold W2n−k,k∈ {1, . . . , n −1}, and hence Mwill be an open part of a tube around this ruled minimal submanifold W2n−k. (If k= 1 then Mwill be an equidistant hypersurface to the ruled minimal hypersurface W2n−1 at distance r.) Let p∈Mand denote by γpthe geodesic determined by the initial conditions γp(0) = pand ˙γp(0) = ξp. For any v∈TpMlet Bvbe the parallel vector field along the geodesic γpsuch that Bv(0) = v, and let ζvbe the Jacobi field along γpwith initial conditions ζv(0) = vand ζ0(0) = −Spv. Here 0denotes covariant derivative along γp. Since ζvis a solution to the differential equation 4ζ00 v+cζv+ 3chζv, J ˙γpiJ˙γp= 0, if v∈Tλi(p) then ζv(t) = fi(t)Bv(t) + hv, Jξigi(t)J˙γp(t), where fi(t) = cosh µt√−c 2¶−2λi √−csinh µt√−c 2¶, gi(t) = µcosh µt√−c 2¶−1¶µ1 + 2 cosh µt√−c 2¶−2λi √−csinh µt√−c 2¶¶. We also define the smooth vector field ηralong Φrby ηr p= ˙γp(r). It is known that ζv(r) = Φr ∗vand ζ0 v(r) = ¯ ∇Φr ∗vηr. We now determine the value of r. Since 0 ≤λ3<√−c/2 we can find a real number r≥0 such that λ3=√−c 2tanh µr√−c 2¶. Let p∈M. We define ui= (Ui)p,i∈ {1,2}. Let v2∈Tλ2(p)ªRu2and vk∈Tλk(p) for 3 ≤k≤g(whenever these spaces are nontrivial). The explicit solution to the Jacobi equation above implies (Φr ∗u1,Φr ∗u2) = (Bu1(r), Bu2(r))D(r), NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 17 Φr ∗v2= 0,Φr ∗v3= sech µr√−c 2¶Bv3(r),Φr ∗v4= 0, where D(t) = µf1(t) + b2 1g1(t)b1b2g2(t) b1b2g1(t)f2(t) + b2 2g2(t)¶. Since det(D(r)) = sech3¡r√−c/2¢we conclude that Φr ∗has constant rank 2n−k (see Theorem 3.12 (bv)-(bvi) for the definition of k). Then, for each point p∈M there exists an open neighborhood Vof psuch that W= Φr(V) is an embedded submanifold of CHn(c) and Φr:V → W is a submersion. (If k= 1, then Φris actually a local diffeomorphism.) Let q= Φr(p)∈ W. The expression above for Φr ∗shows that the tangent space TqWof Wat qis obtained by parallel translation of Ru1⊕Ru2⊕Tλ3(p) along the geodesic γpfrom p=γp(0) to q=γp(r). Therefore, the normal space νqWof W at qis obtained by parallel translation of (ker Φr ∗p)⊕Rξpalong γpfrom p=γp(0) to q=γp(r). The latter is (Tλ2ªRu2)⊕Rξpif g= 3 (see Theorem 3.12 (bvi)), or Tλ4(p)⊕Rξpif g= 4 (see Theorem 3.12 (bv)). In any case, by Theorem 3.12 (bv)- (bvi) it follows that Whas totally real normal bundle of rank k. We have that ηr p=Bξp(r) is a unit normal vector of Wat q. If Srdenotes the shape operator of W, then it is known that Sr ηr pΦr ∗v=−(ζ0 v(r))>, where (·)> denotes orthogonal projection onto the tangent space of W. Using the explicit expression for ζvabove, we get (Sr ηr pBu1(r), Sr ηr pBu2(r)) = (Bu1(r), Bu2(r))C(r),and Sr ηr pBv3(r) = 0 for all v3∈Tλ3(p), where C(r) = −D0(r)D(r)−1. A lengthy and tedious calculation shows that C(r) = √−c 2µ−2b1b2b2 1−b2 2 b2 1−b2 22b1b2¶. Since Jηr p=BJξp(r) = b1Bu1(r) + b2Bu2(r), and BJAp(r) = b2Bu1(r)−b1Bu2(r), the above expression for C(r) implies Sr ηr pBJAp(r) = −√−c 2Jηr p, Sr ηr pJηr p=−√−c 2BJAp(r), and Sr ηr pvanishes on the orthogonal complement of RJηr p⊕RBJAp(r) in TqW. We have that J(νqW ªRηr p) is contained in the parallel translation along γpof Tλ3(p). This follows from Theorem 3.12 (bv)-(bvi) and the fact that νqW ª Rηr p is the parallel translation along γpfrom γp(0) = pto γp(r) = qof Tλ2(p)ªRu2if g= 3, and of Tλ4(p) if g= 4. The linearity of Sr ηp rimplies (13) Sr ηr pJ˜η=−√−c 2hηr p,˜ηiBJAp(r),for all ˜η∈νqW. It follows from the Gauss formula and ¯ ∇J= 0 that Sr ˜ηJηr p=Sr ηr pJ˜η, and hence, Sr ˜ηJηr p= 0 for all ˜η∈νqW ª Rηr p. Let αbe a curve in (Φr)−1({q})∩ V with α(0) = p. Since ηr pand ηr α(t)−hηr α(t), ηr piηr pare perpendicular, Sr ˜ηJηr p= 0, and the linearity of η7→ Sr ηimply 0 = Sr ηr α(t)−hηr α(t),ηr piηr pJηr p=Sr ηr α(t)Jηr p+√−c 2hηr α(t), ηr piBJAp(r), 18 J.C. D´ IAZ-RAMOS AND M. DOM´ INGUEZ-V´ AZQUEZ which together with (13) (with α(t) instead of p) yields −√−c 2hηr α(t), ηr piBJAp(r) = Sr ηr α(t)Jηr p=−√−c 2hηr α(t), ηr piBJAα(t)(r). Since αis arbitrary we get that the map ˜p7→ BJA˜p(r) is constant in the connected component V0of (Φr)−1({q})∩V containing p. Thus it makes sense to define the unit vector z=−BJA˜p(r)∈TqWfor any ˜p∈ V0. We may consider ηras a map from V0to the unit sphere of νqW. The tangent space of V0at pis given by the kernel of Φr ∗p. If v∈ker Φr ∗p, then ηr ∗pv=ζ0 v(r). If g= 3, then v∈ker Φr ∗p=Tλ2(p)ªRu2, and ηr ∗pv=−p−c/2Bv(r). If g= 4, then v∈ker Φr ∗p=Tλ4(p), and ηr ∗pv=−csch(r√−c/2)Bv(r). In any case, we get that ηris a local diffeomorphism from V0into the unit sphere of νqW(note that this is trivial if g= 3 and k= 1). Hence, ηr(V0) is an open subset of the unit sphere of νqW. But since η7→ Sr ηdepends analytically on ηwe conclude Sr ηJη =√−c 2z, Sr ηz=√−c 2Jη, Sr ηv= 0, for all unit η∈νqW, and v∈TqWª(RJη⊕Rz). Therefore, the second fundamental form IIrof Wat qis given by the trivial symmetric bilinear extension of IIr(z, Jη) = (√−c/2)ηfor all η∈νqW. By construction, zdepends smoothly on the point q∈ W and hence gives rise to a vector field Zwhich is tangent to the maximal holomorphic distribution of W. The relation Sr ηJη = (√−c/2)Zensures that Zcan actually be defined on Φr(M), and hence, the second fundamental form of Φr(M) is given by the trivial symmetric bilinear extension of IIr(Z, Jη) = (√−c/2)ηfor all η∈Γ(νΦr(M)). Since Φr(M) has totally real normal bundle of rank kwe conclude from Theorem 2.3, and the remark that follows, that Φr(M) is holomorphically congruent to an open part of the ruled minimal Berndt-Br¨uck submanifold W2n−k. This readily implies that Mis an open part of a tube (an equidistant hypersurface if g= 3 and k= 1) of radius raround the ruled minimal Berndt-Br¨uck submanifold W2n−k. Finally, let us point out that if g= 3 and λ3= 0, then r= 0 and Mis an open part of the ruled minimal hypersurface W2n−1. 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Department of Geometry and Topology, University of Santiago de Compostela, Spain. E-mail address:[email protected] Department of Geometry and Topology, University of Santiago de Compostela, Spain. E-mail address:[email protected]