Isometry invariant Finsler metrics on Hilbert spaces

DOI10.5817/AM2017-3-141zbMATH Open1424.53047arXiv1708.07713MaRDI QIDQ4598940

Eugene Bilokopytov

Publication date: 15 December 2017

Published in: Archivum Mathematicum (Search for Journal in Brave)

Abstract: In this paper we study isometry-invariant Finsler metrics on inner product spaces over $mathbbR$ or $mathbbC$, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most general concept of the Finsler metric is considered without any additional assumptions that are usually built into its definition. However, we present refined versions of the described results for more specific classes of metrics, including the class of Riemannian metrics. Our main result states that for an isometry-invariant Finsler metric the only possible linear maps under which the metric is invariant are scalar multiples of isometries. Furthermore, we characterise the metrics invariant with respect to all linear maps of this type.

Full work available at URL: https://arxiv.org/abs/1708.07713

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zbMATH Keywords

Riemannian metricFinsler metricisometriesunitary invariance

Mathematics Subject Classification ID

Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Global differential geometry of Finsler spaces and generalizations (areal metrics) (53C60) Local differential geometry of Finsler spaces and generalizations (areal metrics) (53B40)

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• MathOverflow

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