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A pseudo W 4 curvature tensor on a Riemannian manifold
Abstract
Recently [Bull. Calcutta Math. Soc. 94, No. 3, 163–166 (2002; Zbl 1028.53016)], the first author introduced the pseudo projective curvature tensor in a Riemannian manifold. In this paper we define a pseudo W 4 curvature tensor on a Riemannian manifold and obtained its several properties.
... The notion of pseudo-W 2 curvature tensor W 2 was given by (Prasad and Maurya [2]) ...
The object of the present paper is to study some curvature condition on LP-Sasakian manifolds which satisfy P. W 2 = 0, W 2. W 2 = 0, L. W 2 = 0, W 2 .L = 0, W 2. W 2 = 0 and W 2 .R = 0.
... whereW 4 is the pseudoW 4 curvature tensor ( [11]). Hence W 4 ([13]) curvature tensor is a particular case of the tensorW 4 . ...
... The pseudo-projective curvature tensor in an n-dimensional Riemannin manifold is given by [1] (4.1) ...
The purpose of the present paper is to study a quarter-symmetric metric connection on a Sasakian manifold. We study the concircular curvature tensor with respect to the quarter-symmetric metric connection satisfying the condition 0 ~). , (~ = Z U R ξ . We also study ξ -pseudo projectively flat and pseudo-projectively flat Sasakian manifold with respect to the quarter-symmetric metric connection. Next, we investigate the nature of semi-symmetric Sasakian manifold admitting quarter-symmetric metric connection. Finally we study quasi-conformally flat Sasakian manifold with respect to the quarter-symmetric metric connection and prove that the manifold is an η -Einstein manifold.
... P is the pseudo-projective curvature tensor given by [2] (1.4) ...
In this paper we study the geometry of trans-Sasakian manifold when it is projective Ricci-semi-symmetric, pseudo-projectively flat and pseudo-projectively semi-symmetric.
Yano and Sawaki(1972) introduced quasi conformal curvature tensor in a Riemannian manifold. Recently one of the author Prasad(2002) investigated pseudo projective curvature tensor in a Riemannian manifold. In this paper we introduced a new curvature tensor named as Pseudo M-projective curvature tensor on a Riemannian manifold. Some properties for PseudoM-projective curvature tensor are investigated. Finally a particular case has been shown. Key words and phrases: Conformal curvature tensor C, projective curvature tensor P, conharmonic curvature tensor H, conciracular curvature tensor V, quasi conformal curvature tensor C, pseudo projective curvature tensor P, quasi concircular curvature tensor Ṽ, M-projective curvature tensor M, pseudo M-projective curvature M and Wi-curvature tensor (i=1, 2,-9).
The object of the present paper is to characterize K-contact manifolds satisfying certain curvature conditions on the Pseudo 4-curvature tensor. AMS2000 Mathematics subject classification 53C15 and 53C25. Keywords-K-contact manifolds, pseudo W 4-flat manifold, ξ-pseudo flat manifold, ϕ-pseudo W 4-flat manifold.
a S(ϕY, ϕZ) =-b[(r+n-1)g(ϕY, ϕZ)-S(ϕY, ϕZ)] + r[ a n−1 + b]g(ϕY, ϕZ). (4.9) By making the use of (1.3) and (1.17) in (4.9), we get (a-b)S(Y,Z) =[−a − b(n − 1) + ar n−1 ]g(Y,Z) +[ ar n−1 − bn] η(Y)η(Z), (4.10) which shows that M n is an η-Einstein manifold. Contracting (4.10), we get b[r-n(n-1)] = 0. (4.11) If b ≠ 0, then from (4.11), we have r = n(n-1). Hence we can state the following theorem: Theorem4.1: Let Mn be an n-dimensional (n>2) ϕ-pseudo W 2-flat LP-Sasakian manifold, then M n is an η-Einstein manifold with the scalar curvature r = n(n-1), provided b ≠ 0. Abstract The object of the present paper is to characterize K-contact manifolds satisfying certain curvature conditions on the Pseudo ̃4-curvature tensor. AMS2000 Mathematics subject classification 53C15 and 53C25. Keywords-K-contact manifolds, pseudo W 4-flat manifold, ξ-pseudo flat manifold, ϕ-pseudo W 4-flat manifold.
Yano and Sawaki,1972) introduced quasi conformal curvature tensor in a Riemannian manifold. Recently one of the author (Prasad,2002) investgated pseudo projective curvature tensore tensor in a Riemannian manifold. In this paper, we defined quasi conharmonic curvature tensor on a Riemannian manifold and obtained its several properties. Finally a particular case has been investigated.
The object of this paper is to study K-contact manifolds with generalized quasi-conformal curvature tensor. We characterized K-contact manifolds satisfying certain curvature conditions on generalized quasi-conformal curvature tensor. Key words and phrases K-contact manifold generalized quasi-conformal curvature tensor, generalized quasi-conformally flat manifold ξ-generalized quasi-conformally flat manifold and irrotational generalized quasi-conformal curvature tensor
De, Shaikh and Sengupta introduced the notion of LP-Sasakian manifolds with coefficient α which generalized the notion of LP-Sasakian manifolds. Recently, Ikawa and his coauthors studied Sasakian manifolds with Lorentzian metric and obtained several results in this manifold. The object of the paper is to steady pseudo flat LP-Sasakian manifolds with coefficient α.
In this paper we introduce a new curvature tensor named as Pseudo
𝑊8 −curvature tensor 𝑊̃8. Some algebraic properties of Pseudo 𝑊8 −curvature tensor 𝑊̃8
are obtained. Further Quadratic killing tensor 𝐴̃ and Quadratic conformal killing tensor 𝐵̃
are considered with respect to Pseudo 𝑊8 −curvature tensor 𝑊̃8. Finally it is proved that
Pseudo 𝑊8 −curvature tensor 𝑊̃8 is conservative if the scalar curvature tensor is constant
under the certain condition.
Yano (1970) investiated a semi-symmetric metric connections in a Riemannian manifold and since then many authors studied this connection. Further Mishra and Pandey (1978) defined a semi-symmetric metric ξ-connection in almost contact manifold and obtained various geometrical properties. Following Mishra and Pandey (1978) we define semi-symmetric metric ξ-connection in Lorentzian Para-Sasakian manifold and study some propeties of curvature tensors.
In this paper we study the geometry of P-Sasakian manifolds under certain conditions like
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