Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions

Abstract

We study Riemannian manifolds M admitting a semi-symmetric metric connection ($) over tilde such that the vector field U is a parallel unit vector field with respect to the Levi-Civita connection del. We prove that ($) over tilde .R = 0 if and only if M is semisymmetric; if ($) over tilde .R = 0 or R.($) over tilde - ($) over tilde .R = 0 or M is semisymmetric and ($) over tilde.($) over tilde = 0, then M is conformally flat and quasi-Einstein. Here R and ($) over tilde denote the curvature tensors of del and ($) over tilde, respectively.