Linear Connections on Normal Almost Contact Manifolds with ...

REMIA 2010Research and Education in Mathematics, Informatics and their ApplicationsDedicated to the 40 th Anniversary of the Faculty of Mathematics & Informatics<strong>Linear</strong> <strong>Connections</strong> on Normal AlmostContact Manifolds withNorden MetricMarta TeofilovaFaculty of Mathematics & InformaticsPlovdiv UniversityDecember, 10 th -12 th , Plovdiv

1. PRELIMINARIESLet (M, ϕ, ξ, η, g) be a (2n + 1)-dimensional almost contact manifoldwith Norden metric (B-metric), i.e. (ϕ, ξ, η) is an almostcontact structure, and g is a pseudo-Riemannian metric, called aNorden metric (B-metric) such that [3]ϕ 2 x = −x + η(x)ξ, η(ξ) = 1,g(ϕx, ϕy) = −g(x, y) + η(x)η(y).The associated metric ˜g of g is defined by˜g(x, y) = g(x, ϕy) + η(x)η(y)and is a Norden metric, too. Both metrics are necessarily of signature(n + 1, n).

Let ∇ be the Levi-Civita connection of g. The fundamental tensor Fis defined byand has the following propertiesF (x, y, z) = g ((∇ x ϕ)y, z)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).The following 1-forms are associated with F :θ(x) = g ij F (e i , e j , x), θ ∗ (x) = g ij F (e i , ϕe j , x),ω(x) = F (ξ, ξ, x).

A classification of the almost contact manifolds with Norden metricis introduced in [3]. Eleven basic classes F i (i = 1, 2, ..., 11) arecharacterized there according to the properties of F .The Nijenhuis tensor N of the almost contact structure (ϕ, ξ, η) isdefined by [5]N(x, y) = ϕ 2 [x, y] + [ϕx, ϕy] − ϕ[ϕx, y] − ϕ[x, ϕy]+ (∇ x η)y.ξ − (∇ y η)x.ξThe almost contact structure in said to be integrable if N = 0. Inthis case the almost contact manifold is called normal [5].In terms of the covariant derivatives of ϕ and η the tensor N isexpressed byN(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 class of the almost contact manifolds with Norden metric satisfyingthe conditions N = 0, dη = 0 isF 1 ⊕ F 2 ⊕ F 4 ⊕ F 5 ⊕ F 6 [3].Analogously to [2], we define an associated tensorÑ of N byÑ(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 class with Ñ = 0 and F (ξ, y, z) = 0 is F 3 ⊕ F 7 [3].

The curvature tensor R of ∇ is defined as usually byR(x, y)z = ∇ x ∇ y z − ∇ y ∇ x z − ∇ [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 theproperties of R, i.e.L(x, y, z, u) = −L(y, x, z, u) = −L(x, y, u, z),L(x, y, z, u) + L(y, z, x, u) + L(z, x, y, u) = 0.A curvature-like tensor L is called a ϕ-Kähler-type tensor ifL(x, y, ϕz, ϕu) = −L(x, y, z, u).

2. LINEAR CONNECTIONS ON ALMOST CONTACTMANIFOLDS WITH NORDEN METRICLet ∇ ′ be a linear connection with deformation tensor Q, i.e.∇ ′ xy = ∇ x y + Q(x, y).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.Theorem 2.1. On an almost contact manifold with Norden metricthere exists a 10-parametric family of almost ϕ-connections ∇ ′defined byg(∇ ′ xy − ∇ x y, z) = Q(x, y, z)with deformation tensor Q given by

Q(x, y, z) = 1 2{F (x, ϕy, z) + F (x, ϕy, ξ)η(z)} − F (x, ϕz, ξ)η(y)+t 1 {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)}+t 2 {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)}+t 3 {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)}+t 4 {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)}+t 5 {F (ϕy, z, ξ) + F (y, ϕz, ξ) − η(y)ω(ϕz)}η(x)+t 6 {F (ϕz, y, ξ) + F (z, ϕy, ξ) − ω(ϕy)η(z)}η(x)+t 7 {F (ϕy, ϕz, ξ) − F (y, z, ξ) + η(y)ω(z)}η(x)+t 8 {F (ϕz, ϕy, ξ) − F (z, y, ξ) + ω(y)η(z)}η(x)+t 9 ω(x)η(y)η(z) + t 10 ω(ϕx)η(y)η(z), t i ∈ R, i = 1, 2, ..., 10.

Corollary 2.1. Let (M, ϕ, ξ, η, g) be a F 1 ⊕ F 2 ⊕ F 4 ⊕ F 5 ⊕ F 6 -manifold. Then, the ϕ-connections ∇ ′ defined in Theorem 2.1 havethe formg(∇ ′ xy − ∇ x y, z) ==2 1 {F (x, ϕy, z) + F (x, ϕy, ξ)η(z)} − F (x, ϕz, ξ)η(y)+s 1 {F (y, x, z) + F (ϕy, ϕx, z)}(2.1)+s 2 {F (y, ϕx, z) − F (ϕy, x, z)}+s 3 F (y, ϕz, ξ)η(x) + s 4 F (y, z, ξ)η(x),where s 1 = t 1 + t 2 , s 2 = t 3 + t 4 , s 3 = 2(t 5 + t 6 ) − t 3 − t 4 ,s 4 = −t 1 − t 2 − 2(t 7 + t 8 ).

Definition 2.2. A linear connection ∇ ′ on an almost contact manifold(M, ϕ, ξ, η) is said to be almost contact if the almost contactstructure (ϕ, ξ, η) is parallel with respect to it, i.e. if∇ ′ ϕ = ∇ ′ ξ = ∇ ′ η = 0.Proposition 2.2. The almost ϕ-connections ∇ ′ defined in Theorem2.1 are almost contact on an almost contact manifold with Nordenmetric and ω = 0.Remark 2.1. The connections defined by (2.1) are almost contacton a F 1 ⊕ F 2 ⊕ F 4 ⊕ F 5 ⊕ F 6 -manifold.

Definition 2.3. A linear connection ∇ ′ on an almost contact manifoldwith Norden metric (M, ϕ, ξ, η, g) is said to be natural if∇ ′ ϕ = ∇ ′ η = ∇ ′ g = 0.Proposition 2.3. Let (M, ϕ, ξ, η, g) be an almost contact manifoldwith Norden metric, and let ∇ ′ be the 10-parametric family of almostϕ-connections defined in Theorem 2.1. Then(i) ∇ ′ are almost contact ifft 1 + t 2 − t 9 = t 3 + t 4 − t 10 = 0;(ii) ∇ ′ are natural ifft 1 + t 2 = t 3 + t 4 = t 5 + t 6 = t 7 + t 8 = t 9 = t 10 = 0.

Theorem 2.4. On an almost contact manifold with Norden metricthere exists a 4-parametric family of natural connections ∇ ′′ definedbyg(∇ ′′ xy − ∇ x y, z) == 1 2{F (x, ϕy, z) + F (x, ϕy, ξ)η(z)} − F (x, ϕz, ξ)η(y)+p 1 {N(y, z, ϕx) + N(ξ, y, ϕx)η(z) + N(z, ξ, ϕx)η(y)}+p 2 {N(z, y, x) + N(y, ξ, x)η(z) + N(ξ, z, x)η(y)}+p 3 {N(ϕ 2 z, y, ξ) + N(z, ξ, ξ)η(y)}η(x)+p 4 {N(y, ϕz, ξ) + N(ϕz, ξ, ξ)η(y)}η(x),where p 1 = t 1 = −t 2 , p 2 = t 3 = −t 4 , p 3 = t 5 = −t 6 , p 4 = t 7 = −t 8 .

The family ∇ ′′ consists of a unique natural connection on normalalmost contact manifolds with Norden metric, namely the connection∇ ′′ xy = ∇ x y + 1 2 {(∇ xϕ)ϕy + (∇ x η)y.ξ} − ∇ x ξ.η(y),which is the well-known canonical connection [1].

Theorem 2.5. On a F 1 ⊕ F 2 ⊕ F 4 ⊕ F 5 ⊕ F 6 -manifold there existsa symmetric almost contact connection ∇ ′ defined by∇ ′ xy = ∇ x y + 1 4{2(∇x ϕ)ϕy + (∇ y ϕ)ϕx − (∇ ϕy ϕ)x+ 2(∇ x η)y.ξ − 3η(x).∇ y ξ − 4η(y).∇ x ξ } .(2.2)Remark 2.2. Let us remark that the connection ∇ ′ defined by (2.2)can be considered as an analogous connection to the well-known Yanoconnection [6,7] on a normal almost contact manifold with Nordenmetric and closed 1-form η.

3. ALMOST CONTACT CONNECTIONS ONF 4 ⊕ F 5 -MANIFOLDSThe classes F 4 and F 5 are defined in [3] by, respectively:F 4 : F (x, y, z) = − θ(ξ)2n{g(ϕx, ϕy)η(z) + g(ϕx, ϕz)η(y)},F 5 : F (x, y, z) = − θ∗ (ξ)2n{g(ϕx, y)η(z) + g(ϕx, z)η(y)}.The subclass of F 4 ⊕ F 5 with closed 1-forms θ and θ ∗ is denoted by(F 4 ⊕ F 5 ) 0 .

The almost contact connections ∇ ′ defined by (2.1) have the followingform on F 4 ⊕ F 5 -manifolds:∇ ′ xy = ∇ x y + 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}(2.3)+ µθ(ξ)−λθ∗ (ξ)2nη(x)ϕy,where λ = s 1 + s 4 , µ = s 3 − s 2 .Let us denote by R ′ the curvature tensor of ∇ ′ , i.e. R ′ (x, y)z =∇ ′ x∇ ′ yz − ∇ ′ y∇ ′ xz − ∇ ′ [x,y] z.

Proposition 3.1. Let (M, ϕ, ξ, η, g) be a (F 4 ⊕F 5 ) 0 -manifold, and∇ ′ be 2-parametric family of almost contact connections defined by(2.3). Then, the curvature tensor R ′ of an arbitrary connection in thefamily (2.3) is of ϕ-Kähler-type and has the formR ′ = R + ξθ(ξ)2n π 5 + ξθ∗ (ξ)2nπ 4 + θ(ξ)24n 2 {π 2 − π 4 }+ θ∗ (ξ) 24n 2 π 1 − θ(ξ)θ∗ (ξ)4n 2 {π 3 − π 5 },where the curvature-like tensors π i (i = 1, 2, 3, 4, 5) are defined by[4]:

π 1 (x, y, z, u) = g(y, z)g(x, u) − g(x, z)g(y, u),π 2 (x, y, z, u) = g(y, ϕz)g(x, ϕu) − g(x, ϕz)g(y, ϕu),π 3 (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),π 4 (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),π 5 (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).Remark 3.1. Let us remark that the result in Proposition 3.1 isobtained in [4] for the curvature tensor of the canonical connectionon a (F 4 ⊕ F 5 ) 0 -manifold, i.e. the connection (2.3) for λ = µ = 0.

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