A note on some generalized curvature tensor

TL;DR

This paper introduces a new class of generalized curvature tensors constructed from the metric and Ricci tensors, exploring their relation to quasi-Einstein and Roter spaces in semi-Riemannian geometry.

Contribution

It defines a novel generalized curvature tensor as a linear combination of Kulkarni-Nomizu products involving the metric, Ricci tensor, and its square, linking to specific geometric spaces.

Findings

Establishes the form of the generalized curvature tensor.

Connects the tensor to quasi-Einstein and Roter spaces.

Provides insights into the structure of semi-Riemannian manifolds.

Abstract

For any semi-Riemannian manifold (M,g) we define some generalized curvature tensor as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.