We will prove that normal complex contact metric manifolds that are Bochner flat must have constant holomorphic sectional curvature 4 and be Kähler. If they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space CP
2n+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

David E. Blair

Martín Molina, Verónica

English

idUS. Depósito de Investigación Universidad de Sevilla

BOCHNER AND CONFORMAL FLATNESS ON NORMAL COMPLEXCONTACT METRIC MANIFOLDSDAVID E. BLAIR AND VERÓNICA MARTÍN-MOLINAAbstract. We will prove that normal complex contact metric manifolds that are Bochnerflat must have constant holomorphic sectional curvature 4 and be Kähler. If they are alsocomplete and simply connected, they must be isometric to the odd-dimensional complexprojective space CP 2n+1(4) with the Fubini-Study metric. On the other hand, it is notpossible for normal complex contact metric manifolds to be conformally flat.1. IntroductionComplex contact manifolds were first introduced by Kobayashi in [6]. However, they werenot as widely studied as real contact manifolds until recently, when more examples have beenpublished. Ishihara and Konishi introduced in [4] a concept of normality, which forced thestructure to be Kählerian and did not include some natural examples like the complex Heisenberggroup. This led Korkmaz to define a weaker version of normality in [7], which included theseexamples as well as the odd-dimensional complex projective spaces and is the notion of normalitythat we use here.In Hermitian geometry, the Bochner tensor plays the role of the Weyl conformal curvaturetensor in real Riemannian geometry. Thus it is natural to study how normal complex contactmetric manifolds are affected by Bochner flatness, i.e. the vanishing of the Bochner conformaltensor of an Hermitian manifold as defined by Tricerri and Vanhecke in [8]. We will prove inSection 3 that this condition means that the manifold must have constant holomorphic sectionalcurvature 4 and be Kähler. Moreover, if they are also complete and simply connected, they mustbe isometric to the odd-dimensional complex projective space. In contrast we show in Section4 that there are no conformally flat normal complex contact metric manifolds.2. PreliminariesWe will first recall the basic concepts and results on complex contact metric manifolds thatwe will use throughout this paper. For more background, see [2] or [7] as a general reference.A complex manifold M with dimCM = 2n+ 1 and complex structure J is a complex contactmanifold if there exists an open covering U = {Oα} of M , such that:(1) On each Oα, there is a holomorphic 1-form ωα with ωα ∧ (dωα)n 6= 0 everywhere,(2) If Oα ∩ Oβ 6= 0, then there is a non-vanishing holomorphic function λαβ in Oα ∩ Oβsuch thatωα = λαβωβ in Oα ∩ Oβ .2000 Mathematics Subject Classification. 53C15.Key words and phrases. Complex contact metric manifold, Bochner tensor, Complex projective space, Weyltensor, Conformally flat.The second author is supported by the FPU scholarship program of the Ministerio de Educación (Spain) andthe PAI group FQM-327 (Junta de Andalućıa, Spain).12 DAVID E. BLAIR AND VERÓNICA MARTÍN-MOLINAOn each Oα, we define Hα = {X ∈ TOα / ωα(X) = 0}. Since the functions λαβ ’sare nonvanishing, Hα = Hβ on Oα ∩ Oβ , so H = ∪Hα is a well-defined, holomorphic,non-integrable subbundle on M , called the horizontal subbundle.From now on, we will suppress the subindexes if Oα is understood.A complex contact manifold M admits a complex almost contact metric structure, i.e. localreal 1-forms u, v = uJ , (1, 1)-tensors G,H = GJ , unit vector fields U and V = −JU and aHermitian metric g such thatH2 = 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 asu′ = au− bv, v′ = bu+ av,G′ = aG− bH, H ′ = bG+ aH,for some functions a, b defined on the overlaps with a2 + b2 = 1. As a result of the aboveformulas, 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 bechosen 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 M has a complex contact metricstructure.On a complex contact metric manifold M , we can write TM = H⊕V, where V is the verticalsubbundle on M , locally spanned by U and 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 metricmanifold M with structure tensors (u, v, U, V,G,H, g) and complex structure J . The followingidentities 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 ofSasakian manifolds, in complex contact geometry we also have the concept of normality. Wewill use the definition that Korkmaz gave in [7], instead of the stronger notion which Ishiharaand 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 3where S and T are (1, 2)-tensors on M defined as followsS(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 ) = (∇GXG)Y − (∇GYG)X −G(∇XG)Y +G(∇YG)Xis the Nijenhuis torsion of G. It was also proved in [7] that:Proposition 2.0.1. Let M be a complex contact metric manifold. Then M is normal if andonly ifg((∇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 covariantderivative of J satisfies:(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 M be a normal complex contact metric manifold. For any horizontal vector field X, theplane section generated by X and Y = aGX + bHX, where a2 + b2 = 1, is called a GH-sectionand 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 sectionspanned by X and Y .If the GH-sectional curvature GHa,b(X) is independent of the choice of a and b, then we willdenote 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 CP 2n+1 with the standard Fubini-Studymetric g of constant holomorphic curvature 4 are examples of normal complex contact metricmanifold 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 M be a normal complex contact metric manifold with constant GH-sectionalcurvature +1 and dσ(U, V ) = −2. Then M is Kähler and has constant holomorphic curvature+4. If, in addition, M in complete and simply connected, then M is isometric to the complexprojective space CP 2n+1 with the Fubini-Study metric.Our conventions for the curvature tensor areR(X,Y )Z = ∇X∇Y Z −∇Y∇X −∇[X,Y ]Z, R(X,Y, Z,W ) = g(R(X,Y )Z,W ).4 DAVID E. BLAIR AND VERÓNICA MARTÍN-MOLINAA number of basic curvature properties of normal complex contact metric manifolds weregiven in [7] and we note the following for our use(7) R(U, V, V, U) = −2dσ(U, V ).For a horizontal vector field X we 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 X and YR(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 andVanhecke in [8]. Define (0, 4)-tensors π1, π2 and L3R by: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 metricmanifolds are in particular), the Bochner conformal tensor is defined asB = R+(18(n+ 1)ψ(ρ∗) +18nϕ(ρ))(R− L3R)+116(2n+ 3)(ϕ+ ψ)(ρ+ 3ρ∗)(R+ L3R) +116(2n− 1)(3ϕ− ψ)(ρ− ρ∗)(R+ L3R)− 132(n+ 1)(2n+ 3)(τ + 3τ∗)(π1 + π2)−132n(2n− 1)(τ − τ∗)(3π1 − π2),where ρ(R), ρ∗(R) are the Ricci tensor and Ricci ∗-tensor associated to the curvature tensorR, which are defined for an arbitrary orthonormal basis {ei} by:ρ(X,Y ) =∑iR(X, ei, ei, Y ), ρ∗(X,Y ) =∑iR(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 B is identically zero. The com-plex projective spaces CP 2n+1(4) mentioned before are Bochner flat because they have constantNORMAL COMPLEX CONTACT METRIC MANIFOLDS 5holomorphic curvature (Theorem 6 of [9]). We will also recall the definition of the Weyl confor-mal tensor. Given a Riemannian manifold M of real dimension m, the Weyl conformal tensoris defined byW (X,Y )Z = R(X,Y )Z +τ(m− 1)(m− 2)(g(Y,Z)X − g(X,Z)Y )(12)+1m− 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 M is said to be conformally flat if it is locally conformally relatedto the Euclidian metric. The following characterizations are well known. If m > 3, M isconformally flat if and only if W = 0. If m = 3, the Weyl conformal tensor is identically zeroand the manifold is conformally flat if and only if the Schouten tensor is a Codazzi tensor (seee.g. [1] for a discussion).3. Normal complex contact metric manifolds which are Bochner flatOn an almost Hermitian manifold we have the following relation between the Ricci tensorand ∗-Ricci tensor, [10] (p. 195)(13) ∇t∇jJ ti −∇j∇tJ ti = (ρjt − ρ∗jt)J ti.Also in [10] (p. 195) one has the result that on a semi-Kähler 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 M suchthat E4n+1 = U and 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∑i=1g((∇EiJ)Ei, Z) = 0and hence that a normal complex contact metric manifold is semi-Kähler. Now rewrite (3.2) asτ − τ∗ =4n+2∑h,i,j=1g((∇EhJ)Ej , Ei)g((∇EjJ)Ei, Eh),which vanishes by virtue of (3). Theorem 3.2. If a normal complex contact metric manifold M is Bochner flat, then it hasconstant holomorphic sectional curvature 4 and is Kähler. If, in addition, M is complete andsimply connected, then it is isometric to the odd-dimensional complex projective space CP 2n+1(4)with the Fubini-Study metric.Proof. By virtue of Theorem 2.1 it is enough to prove that M has constant GH-curvature 1 andsatisfies dσ(U, V ) = −2.We first compute ρ(U,U) and ρ∗(U,U). We have from formulas (7) and (8) thatρ(U,U) =4n+2∑i=1R(U,Ei, Ei, U) =4n∑i=1R(Ei, U, UEi) +R(U, V, V, U)= 4n− 2dσ(U, V ).(16)6 DAVID E. BLAIR AND VERÓNICA MARTÍN-MOLINAAnalogously, we can calculateρ(V, V ) = 4n− 2dσ(U, V ).Now recalling that JU = −V and using equations (9), (2) and (5) (in that order), we obtainρ∗(U,U) =4n+2∑i=1R(U,Ei, JEi, JU) = −4n+2∑i=1R(Ei, U, V, JEi)= −4n∑i=1g(σ(U)Gei + (∇UH)Ei − JEi, JEi) +R(U, V, V, U)=4n∑i=1(g(dσ(Ei, JEi)− 2 + 1)− 2dσ(U, V )= −4n− 2dσ(U, V ) +4n∑i=1dσ(Ei, JEi).(17)Analogously, we also haveρ∗(V, V ) = −4n− 2dσ(U, V ) +4n∑i=1dσ(Ei, JEi).As we have seen the structure is semi-Kähler so we only have the first term on the left handside of (13) which we write in the form∑4n+2i=1 g((∇Ei∇XJ)Ei, Y ). Setting X = Y = U thereis no contribution for Ei = U or Ei = V and we have4n+2∑i=1g((∇Ei∇UJ)Ei, U) =4n∑i=1g(∇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∑i=1dσ(Ei, JEi) = 8nand 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 haveB(U, V, V, U) = R(U, V, V, U) +τ + 3τ∗8(n+ 1)(2n+ 3)(20)+12(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+ τ2(n+ 1).As B(X,U,U,X) = 0 for every X unit and horizontal vector field, using the fact thatρ∗(JX, JX) = ρ∗(X,X) we deduce:0 = 6(2n− 1) + 2(2n− 1)dσ(U, V ) + 2n− 14(n+ 1)τ(23)− 8n2 + 8n− 34nρ(X,X)− 34nρ(JX, JX) + 3ρ∗(X,X).NORMAL COMPLEX CONTACT METRIC MANIFOLDS 7By the first Bianchi Identity and formulas (2) and (9), we know that4n∑i=1R(X, JX,JEi, Ei) = −4n∑i=1R(X, JEi, Ei, JX)−4n∑i=1R(X,Ei, JX, JEi)= 24n+2∑i=1R(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 that4n∑i=1R(X, JX, JEi, Ei) = 2ρ(X,X)− 4+ 44n+2∑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 haveρ∗(X,X) = ρ(X,X)− 2dσ(X, JX) + 4andρ(JX, JX) = ρ(X,X).Using these two equations and (22), formula (23) simplifies to(26) 0 = −2(2n− 3) + 2n+ 14(n+ 1)τ − 6dσ(X, JX)− (2n− 1)ρ(X,X),for every X unit and horizontal.On the other hand, we can deduce from B(X,Y, Y,X) = 0, thatR(X, JX, JX,X) = ρ(X,X) + 2− τ4(n+ 1),R(X,Y, Y,X) = − 22n− 1+τ8(n+ 1)(2n− 1),for all X,Y are unit, horizontal vector fields such that X is orthogonal to Y and JY . The latterequation gives an expression for the GH-curvature:(27) GHa,b(X) = K(X, aGX + bGH) = −22n− 1+τ8(n+ 1)(2n− 1),for every X unit and horizontal.Therefore, equation (10) gives:(28) 6dσ(X, JX) = ρ(X,X) +4n− 62n− 1+ 6− n− 12(n+ 1)(2n− 1)τ.Comparing (26) and (28) we get that(29) ρ(X,X) = − 4n2 − 3n(2n− 1)+4n2 + 2n− 38n(n+ 1)(2n− 1)τ,for every X unit and horizontal. This shows that ρ(X,X) is independent of the vector field X,so dσ(X, JX) does not depend on X either because of (26) or (28). Thus from equation (18)8 DAVID E. BLAIR AND VERÓNICA MARTÍN-MOLINAwe 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+ 14(n+ 1)(2n− 1)τ,for every X unit and horizontal.The formulas (29) and (30) must coincide, which means that τ = 8(n + 1)(2n + 1) (andtherefore 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 to1. The result now follows from Theorem 2.1. 4. Non-existence of normal complex contact metric manifolds which areconformally flatTheorem 4.1. There exist no normal complex contact metric manifolds which are conformallyflat.Proof. The proof is by contradiction. Suppose that M2n+1 is a normal complex contact metricmanifold which is also conformally flat. Then, as dimRM2n+1 = 4n+ 2, we get by (12) thatR(X,Y, Z,W ) = − τ4n(4n+ 1)(g(Y, Z)g(X,W )− g(X,Z)g(Y,W ))(31)− 14n(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 thatR(U, V, V, U) =14n(ρ(U,U) + ρ(V, V ))− τ4n(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 obtaindσ(U, V ) = − 2n2n− 1+τ4(2n− 1)(4n+ 1).Substituting this into (32), we get that(33) ρ(U,U) = ρ(V, V ) =8n22n− 1− τ2(2n− 1)(4n+ 1).On the other hand, we also have from (31) that, for every unit horizontal vector field X onM :R(X,U,U,X) =14n(ρ(X,X) + ρ(U,U))− τ4n(4n+ 1).Applying equations (8) and (33), we obtain that the Ricci tensor satisfies(34) ρ(X,X) = − 4n2n− 1+(4n− 1)τ2(2n− 1)(4n+ 1),for every X unit, horizontal vector field.Finally, if we choose X,Y unit, mutually orthogonal horizontal vector fields in (31), and useformula (34), we have thatR(X,Y, Y,X) = − 22n− 1+τ2(2n− 1)(4n+ 1).In particular, the holomorphic sectional curvature and the GH-sectional curvature of an arbi-trary horizontal vector field X are equal and (6) yields 0 = 3, a contradiction. NORMAL COMPLEX CONTACT METRIC MANIFOLDS 9References[1] Bang, K. and Blair, D.E. : The Schouten tensor and conformally flat manifolds, Topics in DifferentialGeometry, 1–28. Editura Academiei Române, Bucharest (2008)[2] Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkhäuser, Boston (2002)[3] Foreman, B.: Variational Problems on Complex Contact Manifolds with Applications to Twister SpaceTheory. 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, KodaiMath. 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,USAVerónica Mart́ın-Molina, Department of Geometry and Topology, Faculty of Mathematics, Univer-sity of Sevilla, Sevilla, SPAIN.E-mail address: blair@math.msu.edu, veronicamartin@us.es

Bochner and conformal flatness on normal complex contact metric manifolds

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Bochner and conformal flatness on normal complex contact metric manifolds

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