A generalization of a 4-dimensional Einstein manifold

TL;DR

This paper introduces a broader class of 4-dimensional manifolds called weakly Einstein manifolds, extending Einstein manifolds by using curvature identities from the generalized Gauss-Bonnet formula, and provides their characterization.

Contribution

It generalizes the concept of Einstein manifolds in four dimensions by defining and characterizing weakly Einstein manifolds using curvature identities.

Findings

Characterization of weakly Einstein manifolds

Extension of Einstein manifold theory in 4D

Use of generalized Gauss-Bonnet formula

Abstract

A weakly Einstein manifold is a generalization of a 4-dimensional Einstein manifold, which is defined as an application of a curvature identity derived from the generalized Gauss-Bonnet formula for a 4-dimensional compact oriented Riemannian manifold. In this paper, we shall give a characterization of a weakly Einstein manifold.