The Weyl principle on the Finsler frontier
| Autor: | Dmitry Faifman, Thomas Wannerer |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
Polynomial Pure mathematics General Mathematics 010102 general mathematics General Physics and Astronomy Metric Geometry (math.MG) Riemannian manifold 01 natural sciences Mathematics - Metric Geometry Differential Geometry (math.DG) Euclidean geometry Metric (mathematics) FOS: Mathematics Mathematics::Differential Geometry 0101 mathematics Mathematics |
| DOI: | 10.48550/arxiv.1912.09195 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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