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The Weyl principle on the Finsler frontier

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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.

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Acknowledgements

We would like to thank Joe Fu for sparking our interest in this problem through his many inspiring talks, and Semyon Alesker for very helpful comments on the first draft. This work got started while TW was visiting the University of Toronto, during DF’s term as Coxeter Assistant Professor there, and completed during DF’s term as a CRM-ISM postdoctoral fellow in Université de Montréal, McGill and Concordia. The contributions of those institutions are gratefully acknowledged.

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  1. Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07743, Jena, Germany

    Dmitry Faifman

  2. School of Mathematical Sciences, Tel Aviv University, 6997801, Tel Aviv, Israel

    Thomas Wannerer

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  1. Dmitry Faifman
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  2. Thomas Wannerer
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Correspondence to Dmitry Faifman.

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Dmitry Faifman was partially supported by an NSERC Discovery Grant.

Thomas Wannerer supported by DFG Grant WA 3510/1-1.

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Faifman, D., Wannerer, T. The Weyl principle on the Finsler frontier. Sel. Math. New Ser. 27, 27 (2021). https://doi.org/10.1007/s00029-021-00640-7

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