The Weyl principle on the Finsler frontier

TL;DR

This paper explores the applicability of the Weyl principle to Finsler manifolds, revealing that while it generally fails, a weaker form still holds in specific cases, extending the understanding of intrinsic volumes beyond Riemannian geometry.

Contribution

It demonstrates that the Weyl principle does not fully extend to Finsler manifolds but identifies conditions under which a weaker form persists, broadening the scope of intrinsic volume theory.

Findings

Weyl principle generally fails for Finsler manifolds

A weak form of the Weyl principle persists in certain Finsler settings

Extension of intrinsic volume concepts beyond Riemannian geometry

Abstract

Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the coefficients of the tube volume polynomial are intrinsic invariants of the metric. As a consequence, the intrinsic volumes behave naturally under isometric immersions. This phenomenon, subsequently observed in a number of different geometric settings, is commonly referred to as the Weyl principle. In general normed spaces, the Holmes-Thompson intrinsic volumes naturally extend the Euclidean intrinsic volumes. The purpose of this note is to investigate the applicability of the Weyl principle to Finsler manifolds. We show that while in general the Weyl principle fails, a weak form of the principle unexpectedly persists in certain settings.