Study of ???? −projective curvature tensor on a Riemannian manifold

Abstract

The object of the present paper is to study some properties of ‘ ???? − projective’curvature tensor and ‘???? ∗ −projective’ curvature tensor in a Riemannian manifold which have been defined as ′????(????, ????, ????, ????) = ′????(????, ????, ????, ????) − 1 2(???? − 1) [????(????, ????)????????????(????, ????) − ????(????, ????)????????????(????, ????) −????(????, ????)????????????(????, ????) + ????(????, ????)????????????(????, ????)] and ???? ∗(????, ????, ????, ????) = ′????(????, Z, ????, ????) − ଵ ଶ(௡ିଵ) [????(????, ????)????(ℤ, ????) − ????(????, ????)????(ℤ, ????) −????(ℤ, ????)????(????, ????) + ????(ℤ, ????)????(????, ????)], where′???? is the curvature tensor,???? is the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor ???????????? i.e ????(????????, ????) = ????????????(????, ????) and ℤ −tensor of type (0,2).