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On curvature of surfaces immersed in normed spaces

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Abstract

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. The relations of this setting to the field of relative differential geometry are clarified. We obtain characterizations of the Minkowski Gaussian curvature in terms of surface areas, and respective generalizations of the classical theorems of Huber, Willmore, Alexandrov, and Bertrand–Diguet–Puiseux are derived. A generalization of Weyl’s formula for the volume of tubes and some estimates for volumes and areas in terms of curvature are obtained, and in addition we discuss also two-dimensional subcases of the results in more detail.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 24210201, Brazil

    Vitor Balestro & Ralph Teixeira

  2. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Horst Martini

Authors
  1. Vitor Balestro
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  2. Horst Martini
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  3. Ralph Teixeira
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Correspondence to Vitor Balestro.

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Communicated by Andreas Cap.

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Balestro, V., Martini, H. & Teixeira, R. On curvature of surfaces immersed in normed spaces. Monatsh Math 192, 291–309 (2020). https://doi.org/10.1007/s00605-020-01394-8

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