Proceedings of the Estonian Academy of Sciences, vol.57, no.4, pp.210-216, 2008 (SCI-Expanded)
We study Riemannian manifolds M admitting a semi-symmetric metric connection _̃t4 such that the vector field U is a parallel unit vector field with respect to the Levi-Civita connection ▽. We prove that R·R̃ = 0 if and only if M is semisymmetric; if R̃·R = 0 or R̃·R̃ - R̃·R = 0 or M is semisymmetric and R̃·R̃ = 0, then M is conformally flat and quasi-Einstein. Here R and R̃ denote the curvature tensors of ▽ and _̃t4, respectively.