On a Class Almost Contact Manifolds with Norden Metric

REMIA 2010Research and Education in Mathematics, Informatics and their ApplicationsDedicated to the 40 th Anniversary of the Faculty of Mathematics & InformaticsOn a Class Almost Contact Manifoldswith Norden 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 [1]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),ω(x) = F (ξ, ξ, x), ω ∗ = ω ◦ ϕ.θ ∗ (x) = g ij F (e i , ϕe j , x),The corresponding vector field to ω is denoted by Ω, i.e. ω(x) =g(x, Ω).

The Nijenhuis tensor N of the almost contact structure (ϕ, ξ, η) isdefined by [6]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 [6].

A classification of the almost contact manifolds with Norden metricis introduced in [1]. Eleven basic classes F i (i = 1, 2, ..., 11) arecharacterized there according to the properties of F .The classes for which F is expressed explicitly by the other structuraltensors are called main classes.In the present work we focus our attention on the class F 11 given byF 11 :F (x, y, z) = η(x){η(y)ω(z) + η(z)ω(y)}.

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).The Ricci tensor ρ and the scalar curvatures τ and τ ∗ are defined by,respectivelyρ(y, z) = g ij R(e i , y, z, e j ), τ = g ij ρ(e i , e j ), τ ∗ = g ij ρ(e i , ϕe j ).R is called a ϕ-Kähler-type tensor ifR(x, y, ϕz, ϕu) = −R(x, y, z, u).

Let α = {x, y} be a non-degenerate 2-section spanned by the vectorsx, y ∈ T p M, p ∈ M. The sectional curvature of α is defined byk(α; p) =R(x, y, y, x)π 1 (x, y, y, x) ,where π 1 (x, y, z, u) = g(y, z)g(x, u) − g(x, z)g(y, u).In [5] there are introduced the following special sections in T p M:• a ξ-section if α = {x, ξ};• a ϕ-holomorphic section if ϕα = α;• a totally real section if ϕα ⊥ α with respect to g.

The square norms of ∇ϕ, ∇η and ∇ξ are defined by, respectively [3]:)||∇ϕ|| 2 = g ij g ks g((∇ ei ϕ)e k , (∇ ej ϕ)e s ,||∇η|| 2 = ||∇ξ|| 2 = g ij g ks (∇ ei η)e k (∇ ej η)e s .Definition 1.1. An almost contact manifold with Norden metric iscalled isotropic Kählerian if||∇ϕ|| 2 = ||∇η|| 2 = 0 (||∇ϕ|| 2 = ||∇ξ|| 2 = 0).

2. CURVATURE PROPERTIES OF F 11 -MANIFOLDSProposition 2.1. On a F 11 -manifold it is valid||∇ϕ|| 2 = −||N|| 2 = −2||∇η|| 2 = 2ω(Ω).Corollary 2.1. On a F 11 -manifold the following conditions areequivalent:(i) the manifold is isotropic Kählerian;(ii) the vector Ω is isotopic, i.e. ω(Ω) = 0;(iii) the Nijenhuis tensor N is isotropic.

Proposition 2.2. On a F 11 -manifold we haveτ + τ ∗∗ = 2div(ϕΩ) = 2ρ(ξ, ξ),where τ ∗∗ = g is g jk R(e i , e j , ϕe k , ϕe s ).Proposition 2.3 The curvature tensor of a F 11 -manifold with Nordenmetric satisfiesR(x, y, ϕz, ϕu) = −R(x, y, z, u) + ψ 4 (S)(x, y, z, u),where the tensor ψ 4 (S) is defined by [4]ψ 4 (S)(x, y, z, u) = η(y)η(z)S(x, u) − η(x)η(z)S(y, u)+ η(x)η(u)S(y, z) − η(y)η(u)S(x, z).S(x, y) = (∇ x ω)ϕy − ω(ϕx)ω(ϕy).

Proposition 2.4. The curvature tensor of a F 11 -manifold with Nordenmetric is ϕ-Käherian iffwhere ω ∗ = ω ◦ ϕ.(∇ x ω ∗ )y = η(x)η(y)ω(Ω) + ω ∗ (x)ω ∗ (y),

3. AN EXAMPLELet G be a (2n + 1)-dimensional real connected Lie group, and g beits corresponding Lie algebra. If {x 0 , x 1 , ..., x 2n } is a basis of leftinvariantvector fields on G, we define a left-invariant almost contactstructure (ϕ, ξ, η) byϕx i = x i+n , ϕx i+n = −x i , ϕx 0 = 0, i = 1, 2, ..., n,ξ = x 0 , η(x 0 ) = 1, η(x j ) = 0, j = 1, 2, ..., 2n.We also define a left-invariant pseudo-Riemannian metric g on G byg(x 0 , x 0 ) = g(x i , x i ) = −g(x i+n , x i+n ) = 1, i = 1, 2, ..., n,g(x j , x k ) = 0, j ≠ k, j, k = 0, 1, , ..., 2n.Then, (G, ϕ, ξ, η, g) is an almost contact manifold with Norden metric.

Let the Lie algebra g of G be given by the following non-zero commutatorswhere λ i ∈ R.[x i , x 0 ] = λ i x 0 , i = 1, 2, ..., 2n, (3.1)Equalities (3.1) determine a 2n-parametric family of solvable Lie algebras.Further, we study the manifold (G, ϕ, ξ, η, g) with Lie algebra g definedby (3.1).

The components of the Levi-Civita connection:∇ xi x j = ∇ xi ξ = 0, ∇ ξ x i = −λ i ξ, i, j = 1, 2, ..., 2n,∇ ξ ξ = ∑ nk=1 (λ k x k − λ k+n x k+n ).The essential non-zero components of F :F (ξ, ξ, x i ) = ω(x i ) = −λ i+n , F (ξ, ξ, x i+n ) = ω(x i+n ) = λ i ,i = 1, 2, ..., n.Proposition 3.1. The almost contact manifold with Norden metric(G, ϕ, ξ, η, g) defined by (3.1) belongs to the class F 11 .Remark 3.1. The considered manifold has closed 1-forms ω andω ∗ .

Curvature properties• The components of the curvature tensor R:R(x i , ξ, ξ, x j ) = −λ i λ j , i, j = 1, 2, ..., 2n.Because of R(x i , x j , ϕx k , ϕx s ) = 0 for all i, j, k, s = 0, 1, ..., 2n andProposition 2.3 we obtainProposition 3.2. The curvature tensor and the Ricci tensor of theF 11 -manifold (G, ϕ, ξ, η, g) defined by (3.1) have the form, respectivelyR = ψ 4 (S),ρ(x, y) = η(x)η(y)trS + S(x, y),where S(x, y) = (∇ x ω)ϕy − ω(ϕx)ω(ϕy) and trS = div(ϕΩ).

• The essential components of the Ricci tensor ρ:ρ(x i , x j ) = −λ i λ j , i = 1, 2, ..., 2n,ρ(ξ, ξ) = − ∑ )nk=1(λ 2 k − λ2 k+n.• The scalar curvatures τ and τ ∗ :τ = −2n∑k=1( )λ 2 k − λ2 k+n , τ ∗ = −2n∑λ k λ k+n .k=1

• Sectional curvatures:The characteristic 2-sections α ij spanned by the vectors {x i , x j } arethe following:ξ-sections α 0,i (i = 1, 2, ..., 2n)ϕ-holomorphic sections α i,i+n (i = 1, 2, ..., n)the rest α ij are totally real sections.Proposition 3.3. The F 11 -manifold with Norden metric (G, ϕ, ξ, η, g)defined by (3.1) has zero totally real and ϕ-holomorphic sectional curvatures,and its ξ-sectional curvatures are given byλ2 ik(α 0,i ) = − , i = 1, 2, ..., 2n.g(x i , x i )

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