tailieunhanh - A note on Kaehlerian manifolds

The main purpose of the present paper is to study nearly Kaehlerian manifolds. We give the condition for an almost Hermitian manifold to be nearly Kaehlerian. | Turk J Math 30 (2006) , 439 – 445. ¨ ITAK ˙ c TUB A Note on Kaehlerian Manifolds ¨ Tarakcı, A. A. Salimov N. Cengiz, O. Abstract The main purpose of the present paper is to study nearly Kaehlerian manifolds. We give the condition for an almost Hermitian manifold to be nearly Kaehlerian. Key Words: Hybrid tensor, Hermitian manifold, Kaehlerian manifold, Tachibana operator. 1. Introduction Let M be an almost Hermitian manifold with almost complex structure ϕ and hybrid Riemannian metric tensor field g. Then ϕ2 = −I, g(ϕX, ϕY ) = g(X, Y ) (1) for any vector field X and Y on M . We denote by O the operator of covariant differentiation with respect to g in M . If the almost complex structure ϕ of M satisfies (OX ϕ)Y + (OY ϕ)X = 0 for any vector field X and Y on M, then the manifold M is called a nearly Kaehlerian manifold (Tachibana spaces). The condition above reduces to (OX ϕ)X = 0. AMS Mathematics Subject Classification: 53C15, 53C55, 53C56 439 ˙ TARAKCI, SALIMOV CENGIZ, Let N be the Nijenhuis tensor field of ϕ defined by N (X, Y ) = [ϕX, ϕY ] − ϕ[X, ϕY ] − ϕ[ϕX, Y ] − [X, Y ] any vector field X and Y on M . By a simple computation we have N (X, Y ) = −4ϕ(OX ϕ)Y. Proposition: If the Nijenhuis torsion N of a nearly Kaehlerian manifold vanishes, then M is a Keahlarian manifold. We define a Tachibana operator [3] (see also [2, 4]) Φϕ ξ associated with an almost complex structure ϕ and an arbitrary X ∈ =10 (M ) and applied to a tensor ξ ∈ =02 (M ) as Φϕ ξ(X, Z1 , Z2 ) = (LϕX ξ)(Z1 , Z2 ) − LX (ξ ◦ ϕ)(Z1 , Z2 ) (2) +ξ(Z1 , ϕ(LX Z2 )) − ξ(ϕZ1 , LX Z2 ), where LX denotes the operator of Lie derivation with respect to X and (ξ ◦ ϕ)(Z1 , Z2 ) = ξ(ϕZ1 , Z2 ). Expression (2) defines a tensor field Φϕ ξ ∈ =02 (M ) if and only if ξ as a pure tensor [4]. When Φϕ ξ(X, Z1 , Z2 ) = (LϕX ξ)(Z1 , Z2 ) − LX (ξ ◦ ϕ)(Z1 , Z2 ) = 0 (3) for a pure tensor ξ and for any X, Z1 , Z2 ∈ =10 (M ), M being a manifold with almost complex structure ϕ, ξ is said to be almost .