On generalized Roter type manifolds

TL;DR

This paper investigates the geometric properties of generalized Roter type semi-Riemannian manifolds, exploring curvature restrictions, their implications for various symmetry conditions, and providing examples of such manifolds.

Contribution

It introduces the concept of generalized Roter type manifolds, examines their curvature properties, and establishes conditions linking different symmetry structures with new examples.

Findings

Generalized Roter type condition implies various second order restrictions.

Equivalence between local and Ricci symmetries is studied.

Examples confirm the existence of these 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.