Classification of angular curvature measures and a proof of the angularity conjecture

TL;DR

This paper classifies translation-invariant angular smooth curvature measures in Euclidean space and proves the angularity conjecture, showing these measures are preserved under certain geometric transformations on Riemannian manifolds.

Contribution

It provides a complete classification of angular curvature measures and confirms the angularity conjecture, demonstrating their invariance under isometric immersions and Lipschitz-Killing algebra actions.

Findings

Complete classification of angular curvature measures in 

Angular curvature measures are preserved under isometric immersions

Confirmation of the angularity conjecture

Abstract

In this paper angular curvature measures are investigated. Our first result is a complete classification of translation-invariant angular smooth curvature measures on $\mathbb{R}^n$. Subsequently, we use this result to show that the class of angular curvature measures on a Riemannian manifold is preserved by both the pullback by isometric immersions and the action of the Lipschitz-Killing algebra. The latter confirms the angularity conjecture formulated by A. Bernig, J.H.G. Fu, and G. Solanes.