On generalized Roter type manifolds

Abstract

The main object of the present paper is to study the geometric properties of a generalized Roter type semi-Riemannian manifold, which arose in the way of generalization to find the form of the Riemann-Christoffel curvature tensor $R$. Again for a particular curvature restriction on $R$ and the Ricci tensor $S$ there arise two structures, e. g., local symmetry ($\nabla R = 0$) and Ricci symmetry ($\nabla S = 0$); semisymmetry($R\cdot R =0$) and Ricci semisymmetry ($R\cdot S =0$) etc. In differential geometry there is a very natural question about the equivalency of these two structures. In this context it is shown that generalized Roter type condition is a sufficient condition for various important second order restrictions. Some generalizations of Einstein manifolds are also presented here. Finally the proper existence of both type of manifolds are ensured by some suitable examples.