Bochner and conformal flatness on normal complex contact metric manifolds

BOCHNER AND CONFORMAL FLATNESS ON NORMAL COMPLEX CONTACT METRIC MANIFOLDS DAVID E. BLAIR AND VER ´ ONICA MART´ IN-MOLINA Abstract. We will prove that normal complex contact metric manifolds that are Bochner flat must have constant holomorphic sectional curvature 4 and be K¨ahler. If they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space CP2n+1(4) with the Fubini-Study metric. On the other hand, it is not possible for normal complex contact metric manifolds to be conformally flat. 1. Introduction Complex contact manifolds were first introduced by Kobayashi in [6]. However, they were not as widely studied as real contact manifolds until recently, when more examples have been published. Ishihara and Konishi introduced in [4] a concept of normality, which forced the structure to be K¨ahlerian and did not include some natural examples like the complex Heisenberg group. This led Korkmaz to define a weaker version of normality in [7], which included these examples as well as the odd-dimensional complex projective spaces and is the notion of normality that we use here. In Hermitian geometry, the Bochner tensor plays the role of the Weyl conformal curvature tensor in real Riemannian geometry. Thus it is natural to study how normal complex contact metric manifolds are affected by Bochner flatness, i.e. the vanishing of the Bochner conformal tensor of an Hermitian manifold as defined by Tricerri and Vanhecke in [8]. We will prove in Section 3 that this condition means that the manifold must have constant holomorphic sectional curvature 4 and be K¨ahler. Moreover, if they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space. In contrast we show in Section 4 that there are no conformally flat normal complex contact metric manifolds. 2. Preliminaries We will first recall the basic concepts and results on complex contact metric manifolds that we will use throughout this paper. For more background, see [2] or [7] as a general reference. A complex manifold Mwith dimCM= 2n+ 1 and complex structure Jis a complex contact manifold if there exists an open covering U={Oα}of M, such that: (1) On each Oα, there is a holomorphic 1-form ωαwith ωα∧(dωα)n6= 0 everywhere, (2) If Oα∩ Oβ6= 0, then there is a non-vanishing holomorphic function λαβ in Oα∩ Oβ such that ωα=λαβωβin Oα∩ Oβ. 2000 Mathematics Subject Classification. 53C15. Key words and phrases. Complex contact metric manifold, Bochner tensor, Complex projective space, Weyl tensor, Conformally flat. The second author is supported by the FPU scholarship program of the Ministerio de Educaci´on (Spain) and the PAI group FQM-327 (Junta de Andaluc´ıa, Spain). 1 2 DAVID E. BLAIR AND VER´ ONICA MART´ IN-MOLINA On each Oα, we define Hα={X∈TOα/ ωα(X)=0}. Since the functions λαβ’s are nonvanishing, Hα=Hβon Oα∩ Oβ, so H=∪Hαis a well-defined, holomorphic, non-integrable subbundle on M, called the horizontal subbundle. From now on, we will suppress the subindexes if Oαis understood. A complex contact manifold Madmits a complex almost contact metric structure, i.e. local real 1-forms u, v =uJ, (1,1)-tensors G, H =GJ, unit vector fields Uand V=−JU and a Hermitian metric gsuch that H2=G2=−Id +u⊗U+v⊗V, g(GX, Y ) = −g(X, GY ), g(U, X) = u(X), GJ =−JG, GU = 0, u(U)=1, and on the overlaps, the above tensors transform as u0=au −bv, v0=bu +av, G0=aG −bH, H0=bG +aH, for some functions a, b defined on the overlaps with a2+b2= 1. As a result of the above formulas, on a complex almost contact metric manifold M, the following identities also hold (see [4]): HG =−GH =J+u⊗V−v⊗U, JH =−HJ =G, g(HX, Y ) = −g(X, HY ), GV =HU =HV = 0, uG =vG =uH =vH = 0, JV =U, g(U, V )=0. Moreover, given a complex contact manifold, the complex almost contact metric structure can be chosen such that du(X, Y ) = g(X, GY )+(σ∧v)(X, Y ) and dv(X, Y ) = g(X, HY )−(σ∧u)(X, Y ) for some 1-form σ; see [3] or [5]. In this case we say that Mhas a complex contact metric structure. On a complex contact metric manifold M, we can write TM =H ⊕ V, where Vis the vertical subbundle on M, locally spanned by Uand V=−JU, and is usually assumed to be integrable. In this case σ(X) = g(∇XU, V ). From now on, we will work with a complex contact metric manifold Mwith structure tensors (u, v, U, V, G, H, g) and complex structure J. The following identities are established in [7]: ∇UG=σ(U)H, ∇VH=−σ(V)G, dσ(U, X) = v(X)dσ(U, V ), dσ(V, X) = −u(X)dσ(U, V ). In real contact geometry, Sasakian manifolds play an important role. As an analogue of Sasakian manifolds, in complex contact geometry we also have the concept of normality. We will use the definition that Korkmaz gave in [7], instead of the stronger notion which Ishihara and Konishi introduced in [4]. A complex contact metric manifold M is normal if it satisfies the next two conditions: (1) S(X, Y ) = T(X, Y ) = 0 for all X, Y in H, and (2) S(U, X) = T(V, X) = 0 for all X, NORMAL COMPLEX CONTACT METRIC MANIFOLDS 3 where Sand Tare (1,2)-tensors on Mdefined as follows S(X, Y ) = [G, G](X, Y )+2v(Y)HX −2v(X)HY + 2g(X, GY )U−2g(X, HY )V −σ(GX)HY +σ(GY )HX +σ(X)GHY −σ(Y)GHX, T(X, Y ) = [H, H](X, Y )+2u(Y)GX −2u(X)GY + 2g(X, HY )V−2g(X, GY )U +σ(HX)GY −σ(HY )GX −σ(X)HGY +σ(Y)HGX. We recall that [G, G](X, Y )=(∇GX G)Y−(∇GY G)X−G(∇XG)Y+G(∇YG)X is the Nijenhuis torsion of G. It was also proved in [7] that: Proposition 2.0.1. Let Mbe a complex contact metric manifold. Then Mis normal if and only if g((∇XG)Y, Z) = σ(X)g(HY, Z) + v(X)dσ(GZ, GY )−2v(X)g(HGY, Z)(1) −u(Y)g(X, Z)−v(Y)g(JX, Z) + u(Z)g(X, Y )−v(Z)g(X, JY ), g((∇XH)Y, Z) = −σ(X)g(GY, Z)−u(X)dσ(HZ, HY )+2u(X)g(HGY, Z)(2) +u(Y)g(JX, Z)−v(Y)g(X, Z) + u(Z)g(X, JY ) + v(Z)g(X, Y ). As a result of this proposition, on a normal complex contact metric manifold the covariant derivative of Jsatisfies: (3) g((∇XJ)Y, Z) = u(X)(Ω(Z, GY )−2g(HY, Z)) + v(X)(Ω(Z, HY )+2g(GY, Z)). Also on a normal complex contact metric manifold we have: ∇XU=−GX +σ(X)V, ∇XV=−HX −σ(X)U,(4) dσ(GX, GY ) = dσ(HX, HY ) = dσ(Y, X)−2u∧v(Y, X)dσ(U, V ).(5) Let Mbe a normal complex contact metric manifold. For any horizontal vector field X, the plane section generated by Xand Y=aGX +bHX, where a2+b2= 1, is called a GH-section and we define the GH-sectional curvature GHa,b(X) as the curvature of the GH-section, i.e. GHa,b(X) = K(X, aGX +bHX),where K(X, Y ) is the sectional curvature of the plane section spanned by Xand Y. If the GH-sectional curvature GHa,b(X) is independent of the choice of aand b, then we will denote it by GH(X). If it is also independent of the choice of the GH-section at each point, then the holomorphic sectional curvature is (6) K(X, JX) = GH(X)+3, see Proposition 5.2 of [7]. The odd-dimensional complex projective spaces CP2n+1 with the standard Fubini-Study metric gof constant holomorphic curvature 4 are examples of normal complex contact metric manifold and have constant GH-sectional curvature 1 (see [7]). The following result of Korkmaz [7] will be important for our work. Theorem 2.1. Let Mbe a normal complex contact metric manifold with constant GH-sectional curvature +1 and dσ(U, V ) = −2. Then Mis K¨ahler and has constant holomorphic curvature +4. If, in addition, Min complete and simply connected, then Mis isometric to the complex projective space CP2n+1 with the Fubini-Study metric. Our conventions for the curvature tensor are R(X, Y )Z=∇X∇YZ− ∇Y∇X− ∇[X,Y ]Z, R(X, Y, Z, W) = g(R(X, Y )Z, W). 4 DAVID E. BLAIR AND VER´ ONICA MART´ IN-MOLINA A number of basic curvature properties of normal complex contact metric manifolds were given in [7] and we note the following for our use (7) R(U, V, V, U) = −2dσ(U, V ). For a horizontal vector field Xwe have (8) R(X, U)U=X, R(X, V )V=X, (9) R(X, U)V=σ(U)GX + (∇UH)X−JX, R(X, JX, JX, X) + R(X, GX, GX, X) + R(X, HX, HX, X) =−6g(X, X)(dσ(JX, X) + g(X, X)).(10) For horizontal vector fields Xand Y R(X, JX, JY, Y ) = R(X, Y, Y, X) + R(X, JY, JY, X)(11) +4(g(X, GY )dσ(X, HY )−g(X, HY )dσ(X, GY )+2g(X, GY )2+ 2g(X, HY )2). The definition of Bochner tensor of an almost Hermitian manifold was given by Tricerri and Vanhecke in [8]. Define (0,4)-tensors π1,π2and L3Rby: π1(X, Y, Z, W) = g(X, Z)g(Y, W)−g(Y, Z)g(X, W), π2(X, Y, Z, W)=2g(JX, Y )g(JZ, W) + g(JX, Z)g(JY, W)−g(JY, Z)g(JX, W), L3R(X, Y, Z, W) = R(JX, JY, JZ, JW). Given a (0,2)-tensor S, we denote by ϕ(S) and ψ(S): ϕ(S)(X, Y, Z, W) = g(X, Z)S(Y, Z) + g(Y, W)S(X, Z) −g(X, W)S(Y, Z)−g(Y, Z)S(X, W), ψ(S)(X, Y, Z, W)=2g(X, JY )S(Z, JW)+2g(Z, JW)S(X, JY ) +g(X, JZ)S(Y, JW) + g(Y, JW)S(X, JZ) −g(X, JW)S(Y, JZ)−g(Y, JZ)S(X, JW). Given an Hermitian manifold of complex dimension 2n+ 1 (which the complex contact metric manifolds are in particular), the Bochner conformal tensor is defined as B=R+1 8(n+ 1)ψ(ρ∗) + 1 8nϕ(ρ)(R−L3R) +1 16(2n+ 3)(ϕ+ψ)(ρ+ 3ρ∗)(R+L3R) + 1 16(2n−1)(3ϕ−ψ)(ρ−ρ∗)(R+L3R) −1 32(n+ 1)(2n+ 3)(τ+ 3τ∗)(π1+π2)−1 32n(2n−1)(τ−τ∗)(3π1−π2), where ρ(R), ρ∗(R) are the Ricci tensor and Ricci ∗-tensor associated to the curvature tensor R, which are defined for an arbitrary orthonormal basis {ei}by: ρ(X, Y ) = X i R(X, ei, ei, Y ), ρ∗(X, Y ) = X i R(X, ei, Jei, JY ), and τ,τ∗are their associated scalar curvature and ∗-scalar curvature respectively. An almost Hermitian manifold is said to be Bochner flat if Bis identically zero. The complex projective spaces CP2n+1(4) mentioned before are Bochner flat because they have constant NORMAL COMPLEX CONTACT METRIC MANIFOLDS 5 holomorphic curvature (Theorem 6 of [9]). We will also recall the definition of the Weyl conformal tensor. Given a Riemannian manifold Mof real dimension m, the Weyl conformal tensor is defined by W(X, Y )Z=R(X, Y )Z+τ (m−1)(m−2)(g(Y, Z)X−g(X, Z)Y)(12) +1 m−2(g(X, Z)QY −g(Y, Z)QX +ρ(X, Z)Y−ρ(Y, Z)X), for X, Y, Z vector fields on M. A Riemannian manifold Mis said to be conformally flat if it is locally conformally related to the Euclidian metric. The following characterizations are well known. If m > 3, Mis conformally flat if and only if W= 0. If m= 3, the Weyl conformal tensor is identically zero and the manifold is conformally flat if and only if the Schouten tensor is a Codazzi tensor (see e.g. [1] for a discussion). 3. Normal complex contact metric manifolds which are Bochner flat On an almost Hermitian manifold we have the following relation between the Ricci tensor and ∗-Ricci tensor, [10] (p. 195) (13) ∇t∇jJti− ∇j∇tJti= (ρjt −ρ∗ jt)Jti. Also in [10] (p. 195) one has the result that on a semi-K¨ahler manifold (i.e., Ω is coclosed), (14) τ−τ∗= (∇hJji)(∇jJih). In the following, we will always assume {E1, . . . , E4n+2}to be an orthonormal basis of Msuch that E4n+1 =Uand E4n+2 =V. Lemma 3.1. On a normal complex contact metric manifold τ=τ∗. Proof. First note that from (3) we easily have that (15) 4n+2 X i=1 g((∇EiJ)Ei, Z)=0 and hence that a normal complex contact metric manifold is semi-K¨ahler. Now rewrite (3.2) as τ−τ∗= 4n+2 X h,i,j=1 g((∇EhJ)Ej, Ei)g((∇EjJ)Ei, Eh), which vanishes by virtue of (3).  Theorem 3.2. If a normal complex contact metric manifold Mis Bochner flat, then it has constant holomorphic sectional curvature 4and is K¨ahler. If, in addition, Mis complete and simply connected, then it is isometric to the odd-dimensional complex projective space CP2n+1(4) with the Fubini-Study metric. Proof. By virtue of Theorem 2.1 it is enough to prove that Mhas constant GH-curvature 1 and satisfies dσ(U, V ) = −2. We first compute ρ(U, U) and ρ∗(U, U). We have from formulas (7) and (8) that ρ(U, U) = 4n+2 X i=1 R(U, Ei, Ei, U) = 4n X i=1 R(Ei, U, UEi) + R(U, V, V, U) = 4n−2dσ(U, V ).(16) 6 DAVID E. BLAIR AND VER´ ONICA MART´ IN-MOLINA Analogously, we can calculate ρ(V, V )=4n−2dσ(U, V ). Now recalling that JU =−Vand using equations (9), (2) and (5) (in that order), we obtain ρ∗(U, U) = 4n+2 X i=1 R(U, Ei, JEi, JU) = − 4n+2 X i=1 R(Ei, U, V, JEi) =− 4n X i=1 g(σ(U)Gei+ (∇UH)Ei−JEi, JEi) + R(U, V, V, U) = 4n X i=1 (g(dσ(Ei, JEi)−2 + 1) −2dσ(U, V ) =−4n−2dσ(U, V ) + 4n X i=1 dσ(Ei, JEi).(17) Analogously, we also have ρ∗(V, V ) = −4n−2dσ(U, V ) + 4n X i=1 dσ(Ei, JEi). As we have seen the structure is semi-K¨ahler so we only have the first term on the left hand side of (13) which we write in the form P4n+2 i=1 g((∇Ei∇XJ)Ei, Y ). Setting X=Y=Uthere is no contribution for Ei=Uor Ei=Vand we have 4n+2 X i=1 g((∇Ei∇UJ)Ei, U) = 4n X i=1 g(∇Ei((∇UJ)Ei)−(∇UJ)∇EiEi, U), which using (3) and (4) vanishes. Therefore ρ(U, U) = ρ∗(U, U) and using equations (16) and (17) we have (18) 4n X i=1 dσ(Ei, JEi)=8n and in turn (19) ρ(U, U) = ρ(V, V ) = ρ∗(U, U) = ρ∗(V, V )=4n−2dσ(U, V ). Turning to the Bochner tensor, we first have B(U, V, V, U) = R(U, V, V, U) + τ+ 3τ∗ 8(n+ 1)(2n+ 3) (20) +1 2(2n+ 3)−ρ(V, V )−ρ(U, U)−3ρ∗(V, V )−3ρ∗(U, U).(21) Now with B= 0, (7), (19) and τ=τ∗, we have (22) (4n−2)dσ(U, V ) = −16n+τ 2(n+ 1). As B(X, U, U, X) = 0 for every Xunit and horizontal vector field, using the fact that ρ∗(JX, JX) = ρ∗(X, X) we deduce: 0 = 6(2n−1) + 2(2n−1)dσ(U, V ) + 2n−1 4(n+ 1)τ(23) −8n2+ 8n−3 4nρ(X, X)−3 4nρ(JX, JX)+3ρ∗(X, X). NORMAL COMPLEX CONTACT METRIC MANIFOLDS 7 By the first Bianchi Identity and formulas (2) and (9), we know that 4n X i=1 R(X, JX,JEi, Ei) = − 4n X i=1 R(X, JEi, Ei, JX)− 4n X i=1 R(X, Ei, JX, JEi) = 2 4n+2 X i=1 R(X, Ei, JEi, JX)+2R(X, U, V, JX)−2R(X, V, U, JX) = 2ρ∗(X, X)+4−4dσ(X, JX).(24) Using (11) we can also prove that 4n X i=1 R(X, JX, JEi, Ei)=2ρ(X, X)−4 + 4 4n+2 X i=1 (g(X, GEi)dσ(X, HEi)−g(X, HEi)dσ(X, GEi)) + 16 = 2ρ(X, X) + 12 −8dσ(X, JX).(25) Comparing equations (24) and (25), we have ρ∗(X, X) = ρ(X, X)−2dσ(X, JX)+4 and ρ(JX, JX) = ρ(X, X). Using these two equations and (22), formula (23) simplifies to (26) 0 = −2(2n−3) + 2n+ 1 4(n+ 1)τ−6dσ(X, JX)−(2n−1)ρ(X, X), for every Xunit and horizontal. On the other hand, we can deduce from B(X, Y, Y, X) = 0, that R(X, JX, JX, X) = ρ(X, X)+2−τ 4(n+ 1), R(X, Y, Y, X) = −2 2n−1+τ 8(n+ 1)(2n−1), for all X, Y are unit, horizontal vector fields such that Xis orthogonal to Yand JY . The latter equation gives an expression for the GH-curvature: (27) GHa,b(X) = K(X, aGX +bGH) = −2 2n−1+τ 8(n+ 1)(2n−1), for every Xunit and horizontal. Therefore, equation (10) gives: (28) 6dσ(X, JX) = ρ(X, X) + 4n−6 2n−1+ 6 −n−1 2(n+ 1)(2n−1)τ. Comparing (26) and (28) we get that (29) ρ(X, X) = −4n2−3 n(2n−1) +4n2+ 2n−3 8n(n+ 1)(2n−1)τ, for every Xunit and horizontal. This shows that ρ(X, X) is independent of the vector field X, so dσ(X, JX) does not depend on Xeither because of (26) or (28). Thus from equation (18) 8 DAVID E. BLAIR AND VER´ ONICA MART´ IN-MOLINA we obtain that dσ(X, JX) = 2. Substituting in (28), we get another expression for ρ(X, X): (30) ρ(X, X) = −2(2n+ 3) 2n−1+2n+ 1 4(n+ 1)(2n−1)τ, for every Xunit and horizontal. The formulas (29) and (30) must coincide, which means that τ= 8(n+ 1)(2n+ 1) (and therefore ρ(X, X) = 4(n+ 1)). Substituting the expression of the scalar curvature in (22) and (27) we get that dσ(U, V ) = −2 and that the GH-sectional curvature is constant and equal to 1. The result now follows from Theorem 2.1.  4. Non-existence of normal complex contact metric manifolds which are conformally flat Theorem 4.1. There exist no normal complex contact metric manifolds which are conformally flat. Proof. The proof is by contradiction. Suppose that M2n+1 is a normal complex contact metric manifold which is also conformally flat. Then, as dimRM2n+1 = 4n+ 2, we get by (12) that R(X,Y, Z, W) = −τ 4n(4n+ 1)(g(Y, Z)g(X, W)−g(X, Z)g(Y, W))(31) −1 4n(g(X, Z)ρ(Y, W)−g(Y, Z)ρ(X, W) + g(Y, W)ρ(X, Z)−g(X, W)ρ(Y, Z)), for all vector fields X, Y, Z, W on M. In particular, we have that R(U, V, V, U) = 1 4n(ρ(U, U) + ρ(V, V )) −τ 4n(4n+ 1), and by virtue of equation (7) and the fact that (32) ρ(U, U) = ρ(V, V ) = 2(2n−dσ(U, V )), we obtain dσ(U, V ) = −2n 2n−1+τ 4(2n−1)(4n+ 1). Substituting this into (32), we get that (33) ρ(U, U) = ρ(V, V ) = 8n2 2n−1−τ 2(2n−1)(4n+ 1). On the other hand, we also have from (31) that, for every unit horizontal vector field Xon M: R(X, U, U, X) = 1 4n(ρ(X, X) + ρ(U, U)) −τ 4n(4n+ 1). Applying equations (8) and (33), we obtain that the Ricci tensor satisfies (34) ρ(X, X) = −4n 2n−1+(4n−1)τ 2(2n−1)(4n+ 1), for every Xunit, horizontal vector field. Finally, if we choose X, Y unit, mutually orthogonal horizontal vector fields in (31), and use formula (34), we have that R(X, Y, Y, X) = −2 2n−1+τ 2(2n−1)(4n+ 1). In particular, the holomorphic sectional curvature and the GH-sectional curvature of an arbitrary horizontal vector field Xare equal and (6) yields 0 = 3, a contradiction.  NORMAL COMPLEX CONTACT METRIC MANIFOLDS 9 References [1] Bang, K. and Blair, D.E. : The Schouten tensor and conformally flat manifolds, Topics in Differential Geometry, 1–28. Editura Academiei Romˆane, Bucharest (2008) [2] Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkh¨auser, Boston (2002) [3] Foreman, B.: Variational Problems on Complex Contact Manifolds with Applications to Twister Space Theory. Thesis, Michigan State University (1996) [4] Ishihara, S. and Konishi, M.: Complex almost contact manifolds, Kodai Math. J., 3, 385–396 (1980) [5] Ishihara, S. and Konishi, M.: Complex almost contact structures in a complex contact manifold, Kodai Math. J., 5, 30–37 (1982) [6] Kobayashi, S.: Remarks on complex contact manifolds, Proc. Amer. Math., 10, 164–167 (1959) [7] Korkmaz, B.: Normality of complex contact manifolds, Rocky Mountain J. Math., 30, 1343–1380 (2000) [8] Tricerri, F. and Vanhecke, L.: Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc., 267, 365–398 (1981) [9] Vanhecke, L.: The Bochner curvature tensor on almost Hermitian manifolds, Rend. Sem. Mat. Univ. Politec. Torino, 34, 21–38 (1975-76) [10] Yano, K.: Differential Geometry on Complex and Almost Complex Spaces. Pergamon Press, Oxford (1964) David E. Blair, Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Ver´ onica Mart´ ın-Molina, Department of Geometry and Topology, Faculty of Mathematics, University of Sevilla, Sevilla, SPAIN. E-mail address:[email protected], [email protected]