Curvature Measures of Pseudo-Riemannian Manifolds

TL;DR

This paper extends curvature measures from Riemannian to pseudo-Riemannian manifolds, establishing their natural behavior under isometric immersions and generalizing classical theorems like Gauss-Bonnet to these broader contexts.

Contribution

It introduces a family of generalized curvature measures for pseudo-Riemannian manifolds, extending Federer’s Lipschitz-Killing measures and proving their invariance and distributional existence.

Findings

Extended Weyl principle to pseudo-Riemannian manifolds.

Constructed curvature measures invariant under isometric immersions.

Established a Gauss-Bonnet theorem for pseudo-Riemannian manifolds with boundary.

Abstract

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz-Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz-Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern-Gauss-Bonnet integrand. Finally, we deduce a Chern-Gauss-Bonnet theorem for pseudo-Riemannian manifolds with generic boundary.