Isometry invariant Finsler metrics on Hilbert spaces

Eugene Bilokopytov

Archivum Mathematicum (2017)

  • Volume: 053, Issue: 3, page 141-153
  • ISSN: 0044-8753

Abstract

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In this paper we study isometry-invariant Finsler metrics on inner product spaces over ℝ or ℂ , 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 characterize the metrics invariant with respect to all linear maps of this type.

How to cite

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Bilokopytov, Eugene. "Isometry invariant Finsler metrics on Hilbert spaces." Archivum Mathematicum 053.3 (2017): 141-153. <http://eudml.org/doc/294140>.

@article{Bilokopytov2017,
abstract = {In this paper we study isometry-invariant Finsler metrics on inner product spaces over $\mathbb \{R\}$ or $\mathbb \{C\}$, 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 characterize the metrics invariant with respect to all linear maps of this type.},
author = {Bilokopytov, Eugene},
journal = {Archivum Mathematicum},
keywords = {Finsler metric; unitary invariance; isometries; Riemannian metric},
language = {eng},
number = {3},
pages = {141-153},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Isometry invariant Finsler metrics on Hilbert spaces},
url = {http://eudml.org/doc/294140},
volume = {053},
year = {2017},
}

TY - JOUR
AU - Bilokopytov, Eugene
TI - Isometry invariant Finsler metrics on Hilbert spaces
JO - Archivum Mathematicum
PY - 2017
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 053
IS - 3
SP - 141
EP - 153
AB - In this paper we study isometry-invariant Finsler metrics on inner product spaces over $\mathbb {R}$ or $\mathbb {C}$, 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 characterize the metrics invariant with respect to all linear maps of this type.
LA - eng
KW - Finsler metric; unitary invariance; isometries; Riemannian metric
UR - http://eudml.org/doc/294140
ER -

References

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  6. Xia, H., Zhong, Ch., 10.1016/j.difgeo.2015.08.001, Differential Geom. Appl. 43 (2015), 1–20. (2015) Zbl1328.53031MR3421873DOI10.1016/j.difgeo.2015.08.001
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  8. Zhou, L., Spherically symmetric Finsler metrics in R n , Publ. Math. Debrecen 80 (1–2) (2012), 67–77. (2012) MR2920216

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