Certain curvature properties and scalar invariants of the manifolds belonging
to one of the main classes almost contact manifolds with Norden metric are
considered. An example illustrating the obtained results is given and studied

Teofilova, Marta

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arXiv

Marta Teofilova

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On a Class Almost Contact Manifolds with Norden Metric

Certain curvature properties and scalar invariants of the mani-
folds belonging to one of the main classes almost contact manifolds with Norden
metric are considered. An example illustrating the obtained results is given and studied

Bulgarian Digital Mathematics Library at IMI-BAS

ON A CLASS ALMOST CONTACT MANIFOLDSWITH NORDEN METRIC 1Marta TeofilovaAbstract. Certain curvature properties and scalar invariants of the mani-folds belonging to one of the main classes almost contact manifolds with Nordenmetric are considered. An example illustrating the obtained results is given andstudied.Keywords: almost contact manifold, Norden metric, B-metric, isotropicKa¨her manifold.2010 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 (B-metric) in the odd dimensional case.Almost contact manifolds with Norden metric are introduced in [1]. Ele-ven basic classes of these manifolds are characterized there according to theproperties of the covariant derivatives of the almost contact structure.In this work we focus our attention on one of the basic classes almostcontact manifolds with Norden metric, namely the class F11. We study somecurvature properties and relations between certain scalar invariants of themanifolds belonging to this class. In the last section we illustrate the obtainedresults by constructing and studying an example of an F11-manifold on a Liegroup.1. PreliminariesLet M be a (2n + 1)-dimensional smooth manifold, and let (ϕ, ξ, η) bean almost contact structure on M , i.e. ϕ is an endomorphism of the tangentbundle of M , ξ is a vector field, and η is its dual 1-form such that(1.1) ϕ2x = −Id + η ⊗ ξ, η(ξ) = 1.Then, (M,ϕ, ξ, η) is called an almost contact manifold.We equip (M,ϕ, ξ, η) with a compatible pseudo-Riemannian metric g sa-tisfying(1.2) g(ϕx, ϕy) = −g(x, y) + η(x)η(y)1This work is partially supported by The Fund for Scientific Research of the Universityof Plovdiv, Bulgaria, Project RS09-FMI-003.218 Anniversary International Conference REMIA2010for arbitrary x, y in the Lie algebra X(M) of the smooth vector fields on M .Then, g is called a Norden metric (B-metric), and (M,ϕ, ξ, η, g) is called analmost contact manifold with Norden metric.From (1.1), (1.2) 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 oftype (0,3) is defined by(1.3) F (x, y, z) = g ((∇xϕ)y, z)and has the properties(1.4)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).From the last equation and ϕξ = 0 it follows F (x, ξ, ξ) = 0.Let {ei, ξ} (i = 1, 2, ... , 2n) be a basis of the tangent space TpM at anarbitrary point p of M , and gij be the components of the inverse matrix of(gij) with respect to {ei, ξ}. The following 1-forms are associated with F :(1.5)θ(x) = gijF (ei, ej , x), θ∗(x) = gijF (ei, ϕej , x),ω(x) = F (ξ, ξ, x), ω∗ = ω ◦ ϕ.We denote by Ω the vector field corresponding to ω, i.e. ω(x) = g(x,Ω).The Nijenhuis tensor N of the almost contact structure (ϕ, ξ, η) is definedby [6] 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.In terms of the covariant derivatives of ϕ and η the tensor N is expressedas follows(1.6)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).The almost contact structure in said to be integrable if N = 0. In thiscase the almost contact manifold is called normal [6].A classification of the almost contact manifolds with Norden metric isintroduced in [1]. This classification consists of eleven basic classes Fi (i =1, 2, ... , 11) characterized according to the properties of F . The special class F0of the ϕ-Ka¨hler-type almost contact manifolds with Norden metric is given bythe condition F = 0 (∇ϕ = ∇ξ = ∇η = 0). The classes for which F isexpressed explicitly by the other structural tensors are called main classes.10-12 December 2010, Plovdiv, Bulgaria 219In the present work we focus our attention on one of the main classes ofthese manifolds, namely the class F11, which is defined by the characteristiccondition [1](1.7) F (x, y, z) = η(x){η(y)ω(z) + η(z)ω(y)}.By (1.5) and (1.7) we get that on a F11-manifold θ = ω, θ∗ = 0. We alsohave(1.8) (∇xω∗)y = (∇xω)ϕy + η(x)η(y)ω(Ω).The 1-forms ω and ω∗ are said to be closed if dω = dω∗ = 0. Since ∇ issymmetric, necessary and sufficient conditions for ω and ω∗ to be closed are(1.9) (∇xω)y = (∇yω)x, (∇xω)ϕy = (∇yω)ϕx.The curvature tensor R of ∇ is defined as usually by(1.10) R(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).The Ricci tensor ρ and the scalar curvatures τ and τ∗ are defined by, respec-tively(1.11) ρ(y, z) = gijR(ei, y, z, ej), τ = gijρ(ei, ej), τ∗ = gijρ(ei, ϕej).The tensor R is said to be of ϕ-Ka¨hler-type if(1.12) R(x, y, ϕz, ϕu) = −R(x, y, z, u).Let α = {x, y} be a non-degenerate 2-section spanned by the vectorsx, y ∈ TpM , p ∈M . The sectional curvature of α is defined by(1.13) k(α; p) =R(x, y, y, x)pi1(x, y, y, x),where pi1(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 TpM : a ξ-sec-tion if α = {x, ξ}, a ϕ-holomorphic section if ϕα = α and a totally real sectionif ϕα ⊥ α with respect to g.The square norms of ∇ϕ, ∇η and ∇ξ are defined by, respectively [3]:(1.14)||∇ϕ||2 = gijgksg ((∇eiϕ)ek,∇ejϕ)es) ,||∇η||2 = ||∇ξ||2 = gijgks(∇eiη)ek(∇ejη)es.We introduce the notion of an isotropic Ka¨hler-type almost contact ma-nifold with Norden metric analogously to [2].Definition 1.1. An almost contact manifold with Norden metric is calledisotropic Ka¨hlerian if ||∇ϕ||2 = ||∇η||2 = 0 (||∇ϕ||2 = ||∇ξ||2 = 0).220 Anniversary International Conference REMIA20102. Curvature properties of F11-manifoldsIn this section we obtain relations between certain scalar invariants onF11-manifolds with Norden metric and give necessary and sufficient conditionsfor such manifolds to be isotropic Ka¨herian.First, by help of (1.3), (1.4), (1.6), (1.7), (1.14) and direct computationwe obtainProposition 2.1. On a F11-manifold it is valid(2.1) ||∇ϕ||2 = −||N ||2 = −2||∇η||2 = 2ω(Ω).Then, (2.1) and Definition 1.1 yieldCorollary 2.1. On a F11-manifold the following conditions are equiva-lent:(i) the manifold is isotropic Ka¨hlerian;(ii) the vector Ω is isotopic, i.e. ω(Ω) = 0;(iii) the Nijenhuis tensor N is isotropic.It is known that the almost contact structure satisfies the Ricci identity,i.e.(2.2)(∇x∇yϕ)z − (∇y∇xϕ)z = R(x, y)ϕz − ϕR(x, y)z,(∇x∇yη)z − (∇y∇xη)z = −η(R(x, y)z).Then, taking into account the definitions of ϕ, F , and ∇g = 0, the equali-ties (2.2) imply(2.3) (∇xF )(y, z, ϕu)− (∇yF )(x, z, ϕu) = R(x, y, z, u) +R(x, y, ϕz, ϕu)−R(x, y, z, ξ)η(u),(2.4) (∇xF )(y, ϕz, ξ)− (∇yF )(x, ϕz, ξ) = −R(x, y, z, ξ).By (1.3), (2.3) and (2.4) we get(2.5) R(x, y, ϕz, ϕu) = −R(x, y, z, u) + ψ4(S)(x, y, z, u),where the tensor ψ4(S) is defined by [4](2.6)ψ4(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).and(2.7) S(x, y) = (∇xω)ϕy − ω(ϕx)ω(ϕy).Then, the following holdsProposition 2.2. On a F11-manifold we haveτ + τ∗∗ = 2div(ϕΩ) = 2ρ(ξ, ξ),where τ∗∗ = gisgjkR(ei, ej , ϕek, ϕes).10-12 December 2010, Plovdiv, Bulgaria 221Proof. The truthfulness of the statement follows from (1.11) and (2.5) bystraightforward computation.¤Having in mind (1.12) and (2.5), we conclude that the curvature tensor ona F11-manifold is of ϕ-Ka¨her-type if and only if ψ4(S) = 0. Because of (2.6) thelast condition holds true iff S = 0. Then, taking into account (1.8) and (2.7)we proveProposition 2.3. The curvature tensor of a F11-manifold with Nordenmetric is ϕ-Ka¨herian iff(2.8) (∇xω∗)y = η(x)η(y)ω(Ω) + ω∗(x)ω∗(y).The condition (2.8) implies dω∗ = 0, i.e. ω∗ is closed.3. An exampleIn this section we present and study a (2n+ 1)-dimensional example of aF11-manifold constructed on a Lie group.Let G be a (2n + 1)-dimensional real connected Lie group, and g be itscorresponding Lie algebra. If {x0, x1, ... , x2n} is a basis of left-invariant vectorfields on G, we define a left-invariant almost contact structure (ϕ, ξ, η) by(3.1)ϕxi = xi+n, ϕxi+n = −xi, ϕx0 = 0, i = 1, 2, ..., n,ξ = x0, η(x0) = 1, η(xj) = 0, j = 1, 2, ..., 2n.We also define a left-invariant pseudo-Riemannian metric g on G by(3.2)g(x0, x0) = g(xi, xi) = −g(xi+n, xi+n) = 1, i = 1, 2, ... , n,g(xj , xk) = 0, j 6= k, j, k = 0, 1, ... , 2n.Then, according to (1.1) and (1.2), (G,ϕ, ξ, η, g) is an almost contact manifoldwith Norden metric.Let the Lie algebra g of G be given by the following non-zero commutators(3.3) [xi, x0] = λix0, i = 1, 2, ... , 2n,where λi ∈ R. Equalities (3.3) determine a 2n-parametric family of solvableLie algebras.Further, we study the manifold (G,ϕ, ξ, η, g) with Lie algebra g definedby (3.3). The well-known Koszul’s formula for the Levi-Civita connectionof g on G, i.e. the equality2g(∇xixj , xk) = g([xi, xj ], xk) + g([xk, xi], xj) + g([xk, xj ], xi),implies the following components of the Levi-Civita connection:(3.4)∇xixj = ∇xiξ = 0, ∇ξxi = −λiξ, i, j = 1, 2, ... , 2n,∇ξξ =∑nk=1(λkxk − λk+nxk+n).222 Anniversary International Conference REMIA2010Then, by (1.3) and (3.4) we obtain the essential non-zero components of F :(3.5) F (ξ, ξ, xi) = ω(xi) = −λi+n, F (ξ, ξ, xi+n) = ω(xi+n) = λi,for i = 1, 2, ... , n. Hence, by (1.7) and (3.5) we haveProposition 3.1. The almost contact manifold with Norden metric(G,ϕ, ξ, η, g) defined by (3.1), (3.2) and (3.3) belongs to the class F11.Moreover, by (1.9), (3.4) and (3.5) we establish that the considered ma-nifold has closed 1-forms ω and ω∗.Taking into account (3.4) and (1.10) we obtain the essential non-zerocomponents of the curvature tensor as follows(3.6) R(xi, ξ, ξ, xj) = −λiλj , i, j = 1, 2, ... , 2n.By (3.6) it follows that R(xi, xj , ϕxk, ϕxs) = 0 for all i, j, k, s = 0, 1, ..., 2n.Then, according to (2.5) and (2.7) we getProposition 3.2. The curvature tensor and the Ricci tensor of the F11-manifold (G,ϕ, ξ, η, g) defined by (3.1), (3.2) and (3.3) have the form, respec-tivelyR = ψ4(S), ρ(x, y) = η(x)η(y)trS + S(x, y),where S is defined by (2.7) and trS = div(ϕΩ).We compute the essential non-zero components of the Ricci tensor asfollows(3.7)ρ(xi, xj) = −λiλj , i = 1, 2, ... , 2n,ρ(ξ, ξ) = −∑nk=1 (λ2k − λ2k+n) .By (1.11) and (3.7) we obtain the curvatures of the considered manifold(3.8) τ = −2n∑k=1(λ2k − λ2k+n), τ∗ = −2n∑k=1λkλk+n.Let us consider the characteristic 2-sections αij spanned by the vectors{xi, xj}: ξ-sections α0,i (i = 1, 2, ..., 2n), ϕ-holomorphic sections αi,i+n (i =1, 2, ..., n), and the rest are totally real sections. Then, by (1.13), (3.1) and (3.6)it followsProposition 3.3. The F11-manifold with Norden metric (G,ϕ, ξ, η, g) de-fined by (3.1), (3.2) and (3.3) has zero totally real and ϕ-holomorphic sectionalcurvatures, and its ξ-sectional curvatures are given byk(α0,i) = − λ2ig(xi, xi), i = 1, 2, ... , 2n.10-12 December 2010, Plovdiv, Bulgaria 223By (3.2) and (3.5) we obtain the corresponding vector Ω to ω and itssquare norm(3.9) Ω = −n∑k=1(λk+nxk + λkxk+n) , ω(Ω) = −n∑k=1(λ2k − λ2k+n).Then, by (3.9) and Corollary 2.1 we proveProposition 3.4. The F11-manifold with Norden metric (G,ϕ, ξ, η, g) de-fined by (3.1), (3.2) and (3.3) is isotropic Ka¨hlerian iff the conditionn∑k=1(λ2k − λ2k+n)= 0holds.References[1] G. Ganchev, V. Mihova, K. Gribachev: Almost contact ma-nifolds with B-metric, Math. Balk. N.S., 7 (3-4), (1993), 261–276.[2] K. Gribachev, M. Manev, D. Mekerov: A Lie group as a four-dimensional Quasi-Ka¨hler manifold with Norden metric, JP Jour. Geom.Topol., 6(1), (2006), 55–68.[3] S. Kobayashi, K. Nomizu: Foundations of differential geometry, vol. 1,Wiley, New York, (1963).[4] M. Manev: Contactly conformal transformations of general type of al-most contact manifolds with B-metric, Math. Balkanica (N.S.), 11(3-4),(1997), 347-357.[5] G. Nakova, K. Gribachev: Submanifolds of some almost contact man-ifolds with B-metric with codimension two, Math. Balkanica (N.S.), 12(1-2) (1998), 93-108.[6] S. Sasaki: Almost-contact manifolds, Lecture Notes Math., Inst. ToˆhokoUniv. 1-3, (1965), (1967), (1968).Marta TeofilovaFaculty of Mathematics and InformaticsUniversity of Plovdiv236 Bulgaria Blvd.4003 Plovdiv, Bulgariae-mail: marta@uni-plovdiv.bg224 Anniversary International Conference REMIA2010

On a Class Almost Contact Manifolds with Norden Metric

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