Families of linear connections are constructed on almost contact manifolds
with Norden metric. An analogous connection to the symmetric Yano connection is
obtained on a normal almost contact manifold with Norden metric and closed
structural 1-form. The curvature properties of this connection are studied on
two basic classes of normal almost contact manifolds with Norden metric

Teofilova, Marta

English

arXiv

Marta Teofilova

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Linear Connections on Normal Almost Contact Manifolds with Norden Metric

Families of linear connections are constructed on almost con-
tact manifolds with Norden metric. An analogous connection to the symmetric
Yano connection is obtained on a normal almost contact manifold with Norden
metric and closed structural 1-form. The curvature properties of this connec-
tion are studied on two basic classes of normal almost contact manifolds with
Norden metric

Bulgarian Digital Mathematics Library at IMI-BAS

LINEAR CONNECTIONS ON NORMAL ALMOSTCONTACT MANIFOLDS WITH NORDEN METRIC 1Marta TeofilovaAbstract. Families of linear connections are constructed on almost con-tact manifolds with Norden metric. An analogous connection to the symmetricYano connection is obtained on a normal almost contact manifold with Nordenmetric and closed structural 1-form. The curvature properties of this connec-tion are studied on two basic classes of normal almost contact manifolds withNorden metric.Keywords: almost contact manifold, Norden metric, B-metric, linearconnection2010 Mathematics Subject Classification: 53C15, 53C50, 53B30IntroductionThe geometry of the almost contact manifolds with Norden metric (B-metric) is a natural extension of the geometry of the almost complex manifoldswith Norden metric [2] in the case of an odd dimension. A classification of thealmost contact manifolds with Norden metric consisting of eleven basic classesis introduced in [3].An important problem in the geometry of the manifolds equipped withan additional tensor structure and a compatible metric is the study of linearconnections preserving the structures of the manifold. One such connectionon an almost contact manifold with Norden metric, namely the canonicalconnection, is considered in [1, 4].In the present work we construct families of linear connections on analmost contact manifold with Norden metric, which preserve by covariant dif-ferentiation some or all of the structural tensors of the manifold. We obtaina symmetric connection on a normal almost contact manifold with Nordenmetric, which can be considered as an analogue to the well-known Yano con-nection [6, 7].1. PreliminariesLet (M,ϕ, ξ, η, g) be a (2n+1)-dimensional almost contact manifold withNorden metric, i.e. (ϕ, ξ, η) is an almost contact structure: ϕ is an endomor-phism of the tangent bundle of M , ξ is a vector field, η is its dual 1-form,and g is a pseudo-Riemannian metric, called a Norden metric (or a B-metric),1This work is partially supported by The Fund for Scientific Research of the Universityof Plovdiv, Bulgaria, Project RS09-FMI-003.210 Anniversary International Conference REMIA2010such that(1.1)ϕ2x = −x+ η(x)ξ, η(ξ) = 1,g(ϕx, ϕy) = −g(x, y) + η(x)η(y)for all x, y ∈ X(M).From (1.1) it follows ϕξ = 0, η◦ϕ = 0, η(x) = g(x, ξ), g(ϕx, y) = g(x, ϕy).The associated metric g˜ of g is defined by g˜(x, y) = g(x, ϕy)+η(x)η(y) andis a Norden metric, too. Both metrics are necessarily of signature (n+ 1, n).Further, x, y, z, u will stand for arbitrary vector fields in X(M).Let ∇ be the Levi-Civita connection of g. The fundamental tensor F isdefined by(1.2) F (x, y, z) = g ((∇xϕ)y, z)and has the following properties(1.3)F (x, y, z) = F (x, z, y),F (x, ϕy, ϕz) = F (x, y, z)− F (x, ξ, z)η(y)− F (x, y, ξ)η(z).Equalities (1.3) and ϕξ = 0 imply F (x, ξ, ξ) = 0.Let {ei, ξ}, (i = 1, 2, ... , 2n) be a basis of the tangent space TpM atan arbitrary point p of M , and gij be the components of the inverse matrixof (gij) with respect to {ei, ξ}.The following 1-forms are associated with F :θ(x) = gijF (ei, ej , x), θ∗(x) = gijF (ei, ϕej , x), ω(x) = F (ξ, ξ, x).A classification of the almost contact manifolds with Norden metric isintroduced in [3]. Eleven basic classes Fi (i = 1, 2, ... , 11) are characterizedthere according to the properties of F .The Nijenhuis tensor N of the almost contact structure (ϕ, ξ, η) is definedin [5] by N(x, y) = [ϕ,ϕ](x, y) + dη(x, y)ξ, i.e. N(x, y) = ϕ2[x, y] + [ϕx, ϕy]−ϕ[ϕx, y]−ϕ[x, ϕy]+(∇xη)y.ξ−(∇yη)x.ξ. The almost contact structure is saidto be integrable if N = 0. In this case the almost contact manifold is callednormal [5].In terms of the covariant derivatives of ϕ and η the tensor N is expressedby N(x, y) = (∇ϕxϕ)y−(∇ϕyϕ)x−ϕ(∇xϕ)y+ϕ(∇yϕ)x+(∇xη)y.ξ−(∇yη)x.ξ,where (∇xη)y = F (x, ϕy, ξ). Then, according to (1.2), the correspondingtensor of type (0,3), defined by N(x, y, z) = g(N(x, y), z), has the form(1.4)N(x, y, z) = F (ϕx, y, z)− F (ϕy, x, z)− F (x, y, ϕz)+ F (y, x, ϕz) + F (x, ϕy, ξ)η(z)− F (y, ϕx, ξ)η(z).The condition N = 0 and (1.4) imply(1.5) F (x, y, ξ) = F (y, x, ξ), ω = 0.10-12 December 2010, Plovdiv, Bulgaria 211The 1-form η is said to be closed if dη = 0, i.e. if (∇xη)y = (∇yη)x.The class of the almost contact manifolds with Norden metric satisfying theconditions N = 0 and dη = 0 is F1 ⊕F2 ⊕F4 ⊕F5 ⊕F6 [3].Analogously to [2], we define an associated tensor N˜ of N by(1.6)N˜(x, y, z) = F (ϕx, y, z) + F (ϕy, x, z)− F (x, y, ϕz)− F (y, x, ϕz) + F (x, ϕy, ξ)η(z) + F (y, ϕx, ξ)η(z).From N˜ = 0 it follows F (ϕx, ϕy, ξ) = F (y, x, ξ), ω = 0. The class with N˜ = 0is F3 ⊕F7 [3].The curvature tensor R of ∇ is defined as usually byR(x, y)z = ∇x∇yz −∇y∇xz −∇[x,y]z,and its corresponding tensor of type (0,4) is given byR(x, y, z, u) = g(R(x, y)z, u).A tensor L of type (0,4) is said to be curvature-like if it has the propertiesof R, i.e. L(x, y, z, u) = −L(y, x, z, u) = −L(x, y, u, z) and Sx,y,zL(x, y, z, u) = 0(first Bianchi identity), where S is the cyclic sum by three arguments.A curvature-like tensor L is called a ϕ-Ka¨hler-type tensor ifL(x, y, ϕz, ϕu) = −L(x, y, z, u).2. Connections on almost contact manifolds with Norden metricLet ∇′ be a linear connection with deformation tensor Q, i.e. ∇′xy =∇xy +Q(x, y). If we denote Q(x, y, z) = g(Q(x, y), z), then(2.1) g(∇′xy −∇xy, z) = Q(x, y, z).Definition 2.1. A linear connection ∇′ on an almost contact manifoldis called an almost ϕ-connection if ϕ is parallel with respect to ∇′, i.e. if∇′ϕ = 0.Because of (1.2), equality (2.1) and ∇′ϕ = 0 imply the condition for Q(2.2) F (x, y, z) = Q(x, y, ϕz)−Q(x, ϕy, z).Theorem 2.1. On an almost contact manifold with Norden metric thereexists a 10-parametric family of almost ϕ-connections ∇′ of the form (2.1)with deformation tensor Q given by212 Anniversary International Conference REMIA2010(2.3)Q(x, y, z) = 12{F (x, ϕy, z) + F (x, ϕy, ξ)η(z)} − F (x, ϕz, ξ)η(y)+t1{F (y, x, z) + F (ϕy, ϕx, z)− F (y, x, ξ)η(z)− F (ϕy, ϕx, ξ)η(z)−F (y, z, ξ)η(x)− F (ξ, x, z)η(y) + η(x)η(y)ω(z)}+t2{F (z, x, y) + F (ϕz, ϕx, y)− F (z, x, ξ)η(y)− F (ϕz, ϕx, ξ)η(y)−F (z, y, ξ)η(x)− F (ξ, x, y)η(z) + η(x)η(z)ω(y)}+t3{F (y, ϕx, z)− F (ϕy, x, z)− F (y, ϕx, ξ)η(z) + F (ϕy, x, ξ)η(z)−F (y, ϕz, ξ)η(x)− F (ξ, ϕx, z)η(y) + η(x)η(y)ω(ϕz)}+t4{F (z, ϕx, y)− F (ϕz, x, y)− F (z, ϕx, ξ)η(y) + F (ϕz, x, ξ)η(y)−F (z, ϕy, ξ)η(x)− F (ξ, ϕx, y)η(z) + η(x)η(z)ω(ϕy)}+t5{F (ϕy, z, ξ) + F (y, ϕz, ξ)− η(y)ω(ϕz)}η(x)+t6{F (ϕz, y, ξ) + F (z, ϕy, ξ)− ω(ϕy)η(z)}η(x)+t7{F (ϕy, ϕz, ξ)− F (y, z, ξ) + η(y)ω(z)}η(x)+t8{F (ϕz, ϕy, ξ)− F (z, y, ξ) + ω(y)η(z)}η(x)+t9ω(x)η(y)η(z) + t10ω(ϕx)η(y)η(z), ti ∈ R, i = 1, 2, ... , 10.Proof. The proof of the statement follows from (2.2), (2.3) and (1.3) by directverification that ∇′ϕ = 0 for all ti.¤Let N = dη = 0. By (1.4), (1.5) from Theorem 2.1 we obtainCorollary 2.1. Let (M,ϕ, ξ, η, g) be a F1⊕F2⊕F4⊕F5⊕F6-manifold.Then, the deformation tensorQ of the almost ϕ-connections∇′ defined by (2.1)and (2.3) has the form(2.4)Q(x, y, z) = 12{F (x, ϕy, z) + F (x, ϕy, ξ)η(z)} − F (x, ϕz, ξ)η(y)+ s1{F (y, x, z) + F (ϕy, ϕx, z)}+ s2{F (y, ϕx, z)− F (ϕy, x, z)}+ s3F (y, ϕz, ξ)η(x) + s4F (y, z, ξ)η(x),where s1 = t1+t2, s2 = t3+t4, s3 = 2(t5+t6)−t3−t4, s4 = −t1−t2−2(t7+t8).Definition 2.2. A linear connection ∇′ is said to be almost contact if thealmost contact structure (ϕ, ξ, η) is parallel with respect to it, i.e. if ∇′ϕ =∇′ξ = ∇′η = 0.Then, in addition to the condition (2.2), for the deformation tensor Q ofan almost contact connection given by (2.1) we also have(2.5) F (x, ϕy, ξ) = Q(x, y, ξ) = −Q(x, ξ, y).10-12 December 2010, Plovdiv, Bulgaria 213Definition 2.3. A linear connection on an almost contact manifold withNorden metric (M,ϕ, ξ, η, g) is said to be natural if ∇′ϕ = ∇′η = ∇′g = 0.The condition ∇′g = 0 and (2.1) yield(2.6) Q(x, y, z) = −Q(x, z, y).From (1.6) and (2.3) we compute(2.7)Q(x, y, z) +Q(x, z, y) == (t1 + t2){N˜(y, z, ϕx)− N˜(ξ, y, ϕx)η(z)− N˜(ξ, z, ϕx)η(y)}−(t3 + t4){N˜(y, z, x)− N˜(ξ, y, x)η(z)− N˜(ξ, z, x)η(y)}−(t5 + t6){N˜(ϕ2z, y, ξ) + N˜(z, ξ, ξ)η(y)}η(x)+(t7 + t8){N˜(ϕz, y, ξ)− N˜(ϕz, ξ, ξ)η(y)}η(x)+2{t9ω(x) + t10ω(ϕx)}η(y)η(z).By (2.1), (2.3), (2.5), (2.6) and (2.7) we prove the followingProposition 2.1. Let (M,ϕ, ξ, η, g) be an almost contact manifold withNorden metric, and let∇′ be the 10-parametric family of almost ϕ-connectionsdefined by (2.1) and (2.3). Then(i): ∇′ are almost contact iff t1 + t2 − t9 = t3 + t4 − t10 = 0;(ii): ∇′ are natural iff t1+ t2 = t3+ t4 = t5+ t6 = t7+ t8 = t9 = t10 = 0.Taking into account equation (1.4), Theorem 2.1, Proposition 2.1, and byputting p1 = t1 = −t2, p2 = t3 = −t4, p3 = t5 = −t6, p4 = t7 = −t8, weobtainTheorem 2.2. On an almost contact manifold with Norden metric thereexists a 4-parametric family of natural connections ∇′′ defined byg(∇′′xy −∇xy, z) = 12{F (x, ϕy, z) + F (x, ϕy, ξ)η(z)} − F (x, ϕz, ξ)η(y)+ p1{N(y, z, ϕx) +N(ξ, y, ϕx)η(z) +N(z, ξ, ϕx)η(y)}+ p2{N(z, y, x) +N(y, ξ, x)η(z) +N(ξ, z, x)η(y)}+ p3{N(ϕ2z, y, ξ) +N(z, ξ, ξ)η(y)}η(x)+ p4{N(y, ϕz, ξ) +N(ϕz, ξ, ξ)η(y)}η(x).Since N = 0 on a normal almost contact manifold with Norden metric,the family ∇′′ consists of a unique natural connection on such manifolds(2.8) ∇′′xy = ∇xy + 12{(∇xϕ)ϕy + (∇xη)y.ξ} − ∇xξ.η(y),which is the well-known canonical connection [1].Because of Proposition 2.1., (2.7) and the condition N˜ = 0, the connec-tions ∇′ given by (2.1) and (2.3) are natural on a F3⊕F7-manifold iff t1 = −t2and t3 = −t4.214 Anniversary International Conference REMIA2010Let (M, ϕ, ξ, η, g) be in the class F1 ⊕ F2 ⊕ F4 ⊕ F5 ⊕ F6. Then, N =dη = 0 and hence the torsion tensor T of the 4-parametric family of almostϕ-connections ∇′ defined by (2.1) and (2.4) has the form(2.9)T (x, y, z) = s1 {F (y, x, z)− F (x, y, z) + F (ϕy, ϕx, z)− F (ϕx, ϕy, z)}+ 1−4s22 {F (x, ϕy, z)− F (y, ϕx, z)}− s4{F (x, z, ξ)η(y)− F (y, z, ξ)η(x)}− (1− s2 + s3){F (x, ϕz, ξ)η(y)− F (y, ϕz, ξ)η(x)}.Then, from (2.9) we derive that T = 0 if and only if s1 = s4 = 0, s2 = 14 ,s3 = −34 . By this way we proveTheorem 2.3. On a F1 ⊕ F2 ⊕ F4 ⊕ F5 ⊕ F6-manifold there exists asymmetric almost ϕ-connection ∇′ defined by(2.10)∇′xy = ∇xy + 14 {2(∇xϕ)ϕy + (∇yϕ)ϕx− (∇ϕyϕ)x+ 2(∇xη)y.ξ − 3η(x).∇yξ − 4η(y).∇xξ} .Let us remark that the connection (2.10) can be considered as an analo-gous connection to the well-known Yano connection [6, 7] on a normal almostcontact manifold with Norden metric and closed 1-form η.3. Connections on F4 ⊕F5-manifoldsIn this section we study the curvature properties of the 4-parametric fam-ily of almost ϕ-connections ∇′ given by (2.1) and (2.4) on two of the ba-sic classes normal almost contact manifolds with Norden metric, namely theclasses F4 and F5. These classes are defined in [3] by the following character-istic conditions for F , respectively:(3.1) F4 : F (x, y, z) = − θ(ξ)2n {g(ϕx, ϕy)η(z) + g(ϕx, ϕz)η(y)},(3.2) F5 : F (x, y, z) = − θ∗(ξ)2n {g(ϕx, y)η(z) + g(ϕx, z)η(y)}.The subclasses of F4 and F5 with closed 1-form θ and θ∗, respectively, aredenoted by F04 and F05 . Then, it is easy to prove that on a F04 ⊕F05 -manifoldit is valid:(3.3) xθ(ξ) = ξθ(ξ)η(x), xθ∗(ξ) = ξθ∗(ξ)η(x).Taking into consideration (3.1) and (3.2), from (2.1) and (2.4) we obtainProposition 3.1. Let (M, ϕ, ξ, η, g) be a F4 ⊕ F5-manifold. Then, theconnections ∇′ defined by (2.1) and (2.4) are given by(3.4)∇′xy = ∇xy + θ(ξ)2n {g(x, ϕy)ξ − η(y)ϕx}+ θ∗(ξ)2n {g(x, y)ξ − η(y)x}+ λθ(ξ)+µθ∗(ξ)2n {η(x)y − η(x)η(y)ξ}+ µθ(ξ)−λθ∗(ξ)2n η(x)ϕy,where λ = s1 + s4, µ = s3 − s2.10-12 December 2010, Plovdiv, Bulgaria 215The Yano-type connection (2.10) is obtained from (3.4) for λ = 0, µ = −1.Let us denote by R′ the curvature tensor of ∇′, i.e. R′(x, y)z = ∇′x∇′yz−∇′y∇′xz −∇′[x,y]z. The corresponding tensor of type (0,4) with respect to g isdefined by R′(x, y, z, u) = g(R′(x, y)z, u). Then, it is validProposition 3.2. Let (M, ϕ, ξ, η, g) be a F04 ⊕ F05 -manifold, and ∇′be 2-parametric family of almost ϕ-connections defined by (3.4). Then, thecurvature tensor R′ of an arbitrary connection in the family (3.4) is of ϕ-Ka¨hler-type and has the form(3.5)R′ = R+ ξθ(ξ)2n pi5 +ξθ∗(ξ)2n pi4 +θ(ξ)24n2{pi2 − pi4}+ θ∗(ξ)24n2pi1 − θ(ξ)θ∗(ξ)4n2{pi3 − pi5},where the curvature-like tensors pii (i = 1, 2, 3, 4, 5) are defined by [4]:(3.6)pi1(x, y, z, u) = g(y, z)g(x, u)− g(x, z)g(y, u),pi2(x, y, z, u) = g(y, ϕz)g(x, ϕu)− g(x, ϕz)g(y, ϕu),pi3(x, y, z, u) = −g(y, z)g(x, ϕu) + g(x, z)g(y, ϕu)− g(x, u)g(y, ϕz) + g(y, u)g(x, ϕz),pi4(x, y, z, u) = g(y, z)η(x)η(u)− g(x, z)η(y)η(u)+ g(x, u)η(y)η(z)− g(y, u)η(x)η(z),pi5(x, y, z, u) = g(y, ϕz)η(x)η(u)− g(x, ϕz)η(y)η(u)+ g(x, ϕu)η(y)η(z)− g(y, ϕu)η(x)η(z).Proof. It is known that the curvature tensors of two linear connections relatedby an equation of the form (2.1) satisfy(3.7)g(R′(x, y)z, u) = R(x, y, z, u) + (∇xQ)(y, z, u)− (∇yQ)(x, z, u)+Q(x,Q(y, z), u)−Q(y,Q(x, z), u).Then, (3.5) follows from (3.7), (3.4), (3.3) and (3.6) by straightforward com-putation. ¤Let us remark that (3.5) is obtained in [4] for the curvature tensor of thecanonical connection (2.8) on a F04 ⊕ F05 -manifold, i.e. the connection (3.4)for λ = µ = 0.References[1] V. Alexiev, G. Ganchev: Canonical connection on a conformal almostcontact metric manifold, Ann. Univ. Sofia-Math. 81, (1987), 29–38.[2] G. Ganchev, A. Borisov: Note on the almost complex manifolds witha Norden metric, Compt. Rend. Acad. Bulg. Sci. 39(5), (1986), 31–34.[3] G. Ganchev, V. Mihova, K. Gribachev: Almost contact manifoldswith B-metric, Math. Balk. N.S. 7(3-4) (1993), 261–276.216 Anniversary International Conference REMIA2010[4] M. Manev: On the curvature properties of almost contact B-metrichypersurfaces of Kaehlerian manifolds with B-metric, Plovdiv Univ. Sci.Works – Math. 33(3), (2001), 61-72.[5] S. Sasaki: Almost-contact manifolds, Lecture Notes Math., Inst. ToˆhokoUniv. 1-3, 1965, 1967, 1968.[6] K. Yano: Affine connections in an almost product space, Kodai Math.Semin. Rep. 11(1), (1959), 1–24.[7] K. Yano: Differential geometry on complex and almost complex spaces,Pure and Applied Math., vol. 49, Pergamon Press Book, New York, 1965.Marta TeofilovaFaculty of Mathematics and InformaticsUniversity of Plovdiv236 Bulgaria Blvd.4003 Plovdiv, Bulgariae-mail: marta@uni-plovdiv.bg

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Linear Connections on Normal Almost Contact Manifolds with Norden Metric

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