Abstract
The purpose of the present paper is to study some properties of the Projective curvature tensor with respect to Zamkovoy connection in Lorentzian Para Sasakian manifold(or,LP-Sasakian manifold)'And we have studied some results in Lorentzian Para-Sasakian manifold with the help of Zamkovoy connection and Projective curvature tensor.Also we discussed the LP-Sasakian manifold satisfying P*(ξ,U)∘W₀*=0,P*(ξ,U)∘W₂*=0 , where W₀*,W₂* and P* are W₀,W₂ and Projective curvature tensors with respect to Zamkovoy connection.
Highlights
Let us introduced a symmetric (0, 2) tensor field ω such that ω (X, Y ) = g(X, φY )
An n−dimensional LP-Sasakian manifold M (n > 3) is locally projectively φ-symmetric with respect to Zamkovoy connection if and only if it is so with respect to the Levi-Civita connection, provided trace (φ) = 0
Summary
Due to [4], the Projective curvature tensor P of rank four for an n-dimensional Riemannian Manifold M is given by. For an n-dimensional almost contact metric manifold M equipped with an almost contact metric structure (φ, ξ, η, g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a Riemannian metric g, the Zamkovoy connection (∇∗) in terms of Levi-Civita connection (∇) is given by. In a LP-Sasakian manifold M of dimension (n > 2), the Projective curvature tensor P, W0 Curvature tensor [10], W2−Curvature tensor [12] with respect to the Levi-Civita connection are given by. In section (3), we have discussed LP-Sasakian manifold admitting Zamkovoy connection ∇∗ and obtain curvature tensor R∗, Ricci tensor S∗, Scalar curvature tensor r∗, in LP-Sasakian manifold.
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