On equivalency of various geometric structures

TL;DR

This paper investigates the equivalence of different geometric structures formed by imposing the same curvature restrictions on various curvature tensors, unifying them through a combined tensor.

Contribution

It introduces a new tensor that unifies various curvature tensors, facilitating the study of their equivalence under similar restrictions.

Findings

Unified tensor representation of curvature tensors

Conditions for equivalency of geometric structures

Simplification of comparing different curvature restrictions

Abstract

In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structure given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann-Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases.