The Weyl principle on the Finsler frontier

Dmitry Faifman, Thomas Wannerer

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number27
JournalSelecta Mathematica, New Series
Volume27
Issue number2
DOIs
StatePublished - May 2021
Externally publishedYes

Keywords

  • Cosine transform
  • Finsler manifolds
  • Holmes–Thompson volume
  • Intrinsic volumes
  • Lipschitz–Killing curvatures
  • Minkowski geometry
  • Quermassintegrals
  • Valuations on manifolds
  • Weyl tube formula

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