On weak symmetries of trans-Sasakian manifolds

The present paper deals with weakly symmetric and weakly Ricci symmetric trans-Sasakian manifolds. The existence of weakly Ricci symmetric trans-Sasakian manifolds is ensured by an example.


INTRODUCTION
As a proper generalization of pseudosymmetric manifolds by Chaki [3], in 1989 Tamássy and Binh [14] introduced the notion of weakly symmetric manifolds.A non-flat Riemannian manifold (M n , g) (n > 2) is called weakly symmetric if its curvature tensor R of type (0, 4) satisfies the condition for all vector fields X,Y, Z,U,V ∈ χ(M n ), where A, B,C, D and E are 1-forms (not simultaneously zero) and ∇ denotes the operator of covariant differentiation with respect to the Riemannian metric g.The 1-forms are called the associated 1-forms of the manifold and an n-dimensional manifold of this kind is denoted by (W S) n .If in (1.1) the 1-form A is replaced by 2A and E is replaced by A, then a (W S) n reduces to the notion of generalized pseudosymmetric manifold by Chaki [4].In 1999 De and Bandyopadhyay [6] studied a (W S) n and proved that in such a manifold the associated 1-forms B = C and D = E. Hence (1.1) reduces to the following: In 1993 Tamássy and Binh [15] introduced the notion of weakly Ricci symmetric manifolds.A Riemannian manifold (M n , g) (n > 2) is called weakly Ricci symmetric if its Ricci tensor S of type (0, 2) is not identically zero and satisfies the condition (∇ X S)(Y, Z) = A(X)S(Y, Z) + B(Y )S(X, Z) +C(Z)S(Y, X), (1.3) where A, B, C are three non-zero 1-forms, called the associated 1-forms of the manifold, and ∇ denotes the operator of covariant differentiation with respect to the metric tensor g.Such an n-dimensional manifold is denoted by (W RS) n .As an equivalent notion of (W RS) n , Chaki and Koley [5] introduced the notion of generalized pseudo Ricci symmetric manifold.If in (1.3) the 1-form A is replaced by 2A then the definition of (W RS) n reduces to that of generalized pseudo Ricci symmetric manifold by Chaki and Koley.Especially, if A = B = C = 0, then a (W RS) n reduces to Ricci-symmetric and if B = C = 0, then it reduces to Ricci recurrent.
The object of the present paper is to study weakly symmetric and weakly Ricci symmetric trans-Sasakian manifolds.Section 2 deals with preliminaries of trans-Sasakian manifolds.Tamássy and Binh [15] studied weakly symmetric and weakly Ricci symmetric Sasakian manifolds and proved that in such a manifold the sum of the associated 1-forms vanishes everywhere.Subsequently in [7] De et al. considered weakly symmetric and weakly Ricci symmetric K-contact manifolds.Also De et al. [8] studied weakly symmetric and weakly Ricci symmetric contact metric manifolds with a nullity condition.Again Özgür [12] studied weakly symmetric and weakly Ricci symmetric Kenmotsu manifolds and proved that in such a manifold the sum of the associated 1-forms is zero everywhere and hence such a manifold does not exist unless the sum of the associated 1-forms is everywhere zero.However, in Section 3 of the paper it is proved that the sum of the associated 1-forms of a weakly symmetric trans-Sasakian manifold of non-vanishing ξ -sectional curvature is non-zero everywhere and hence such a structure exists.In Section 4 we study weakly Ricci symmetric trans-Sasakian manifolds and prove that in such a structure, with non-vanishing ξ -sectional curvature, the sum of the associated 1-forms is non-vanishing everywhere and consequently such a structure exists.Finally, Section 5 deals with a concrete example of weakly Ricci symmetric trans-Sasakian manifold that is neither Ricci symmetric nor Ricci-recurrent.

TRANS-SASAKIAN MANIFOLDS
A (2n + 1)-dimensional smooth manifold M is said to be an almost contact metric manifold [1] if it admits a (1, 1) tensor field φ , a vector field ξ , a 1-form η, and a Riemannian metric g, which satisfy for all vector fields X,Y on M.
An almost contact metric manifold M 2n+1 (φ , ξ , η, g) is said to be trans-Sasakian manifold [11] if (M × R, J, G) belongs to the class W 4 of the Hermitian manifolds, where J is the almost complex structure on M × R defined by for any vector field Z on M and smooth function f on M × R and G is the product metric on M × R.This may be stated by the condition [2] where α, β are smooth functions on M and such a structure is said to be the trans-Sasakian structure of type (α, β ).From (2.4) it follows that In a trans-Sasakian manifold M 2n+1 (φ , ξ , η, g) the following relations hold [9]: ) (2.10) ) where R is the curvature tensor of type (1, 3) of the manifold and Q is the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor S, that is, g(QX,Y ) = S(X,Y ) for any vector fields X, Y on M. The ξ -sectional curvature K(ξ , X) = g(R(ξ , X)ξ , X) for a unit vector field X orthogonal to ξ plays an important role in the study of an almost contact metric manifold.Throughout the paper we consider a trans-Sasakian manifold of non-vanishing ξ -sectional curvature.
Corollary 3.5.If a β -Kenmotsu manifold is weakly symmetric, then the sum of the 1-forms, i.e.A + B + D, is given by
Theorem 4.1.In a weakly Ricci symmetric trans-Sasakian manifold (M 2n+1 , g) of non-vanishing ξsectional curvature the following relations hold: where r is the scalar curvature of the manifold, div denotes the divergence, ρ 1 , ρ 2 being the associated vector fields corresponding to the 1-forms A and B, respectively, and ψ = tr(Qφ ).
Let ∇ be the Levi-Civita connection with respect to the Riemannian metric g and R be the curvature tensor of g of type (1,3).Then we have Taking E 3 = ξ and using Koszul formula for the Riemannian metric g, we can easily calculate From the above it can be easily seen that (φ , ξ , η, g) is a trans-Sasakian structure on M. Consequently M 3 (φ , ξ , η, g) is a trans-Sasakian manifold with α = − 1 2 e 2z = 0 and β = −1.Using the above relations, we can easily calculate the non-vanishing components of the curvature tensor as follows: and the components which can be obtained from these by the symmetry properties.
Using the components of the curvature tensor, we can easily calculate the non-vanishing components of the Ricci tensor S and its covariant derivatives as follows: Since {E 1 , E 2 , E 3 } is an orthonormal basis of (M 3 , g), any vector X and Y can be written as With these 1-forms the manifold under consideration is a weakly Ricci symmetric trans-Sasakian manifold.This leads to the following: Theorem 5.1.There exists a trans-Sasakian manifold (M 3 , g) which is weakly Ricci symmetric but neither Ricci symmetric nor Ricci-recurrent.
Remark.Özgür [12] proved that in a weakly Ricci symmetric Kenmotsu manifold the sum of its associated 1-forms is zero everywhere, but in a weakly Ricci symmetric trans-Sasakian manifold the sum of its associated 1-forms is non-zero everywhere unless the manifold is of non-vanishing ξ -sectional curvature.