Uniqueness of curvature measures in pseudo-Riemannian geometry

TL;DR

This paper proves that Lipschitz-Killing curvature measures in pseudo-Riemannian geometry are uniquely determined by their invariance under isometric embeddings, and applies this to classify measures on isotropic space forms.

Contribution

It establishes the uniqueness of Lipschitz-Killing curvature measures based on invariance, and classifies invariant measures on isotropic pseudo-Riemannian space forms.

Findings

Lipschitz-Killing curvature measures are uniquely characterized by the Weyl principle.

A Künneth-type formula for these measures is proven.

Invariant generalized valuations and curvature measures are classified on isotropic pseudo-Riemannian space forms.

Abstract

The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a K\"unneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.