Universal curvature identities III

TL;DR

This paper investigates universal curvature identities in pseudo-Riemannian manifolds with boundary, deriving Euler-Lagrange equations related to the Chern-Gauss-Bonnet formula that depend only on curvature and second fundamental form.

Contribution

It generalizes Berger's conjecture by showing that the Euler-Lagrange equations are expressed solely in terms of curvature and second fundamental form, without covariant derivatives.

Findings

Euler-Lagrange equations depend only on curvature and second fundamental form

Generalization of Berger's conjecture to manifolds with boundary

Identification of universal curvature identities in pseudo-Riemannian geometry

Abstract

We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that they are given solely in terms of curvature {and the second fundamental form and do not involve covariant derivatives thus generalizing a conjecture of Berger to this context.