Some semisymmetry conditions on Riemannian manifolds

Abstract

We study a Riemannian manifold M admitting a semisymmetric metric connection ∇ such that the vector field U is a parallel unit vector field with respect to the Levi-Civita connection ∇. Firstly, we show that if M is projectively flat with respect to the semisymmetric metric connection ∇ then M is a quasi-Einstein manifold. Also we prove that if R⋅P=0 if and only if M is projectively semisymmetric; if P⋅R=0 or R⋅P-P⋅R=0 then M is conformally flat and quasi-Einstein manifold. Here R, P and P denote Riemannian curvature tensor, the projective curvature tensor of ∇ and the projective curvature tensor of ∇, respectively.