Universal curvature identities II

TL;DR

This paper demonstrates that universal curvature identities valid in Riemannian geometry extend to pseudo-Riemannian manifolds, and studies the associated Euler-Lagrange equations, showing they depend only on curvature in both settings.

Contribution

It extends known curvature identities and properties of Euler-Lagrange equations from Riemannian to pseudo-Riemannian geometry.

Findings

Universal curvature identities extend to pseudo-Riemannian manifolds.

Euler-Lagrange equations depend solely on curvature, not derivatives.

The Euh-Park-Sekigawa identity holds in pseudo-Riemannian setting.

Abstract

We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in the Riemannian setting, they are given solely in terms of curvature (and not in terms of covariant derivatives of curvature) even in the pseudo-Riemannian setting.