A Curvature identity on a 4-dimensional Riemannian manifold

TL;DR

This paper derives a curvature identity for 4-dimensional Riemannian manifolds from the generalized Gauss-Bonnet formula, applicable to both compact and non-compact cases, with various applications demonstrated.

Contribution

It introduces a new curvature identity for 4D Riemannian manifolds derived from the generalized Gauss-Bonnet formula, extending its applicability.

Findings

Curvature identity holds on non-compact 4D manifolds

Derived from generalized Gauss-Bonnet formula

Applications demonstrate utility of the identity

Abstract

We give a curvature identity derived from the generalized Gauss-Bonnet formula for 4-dimensional compact oriented Riemannian manifolds. We prove that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact. We also provide some applications of the identity.