Isometry invariant Finsler metrics on Hilbert Spaces

TL;DR

This paper characterizes all isometry-invariant Finsler metrics on inner product spaces, providing a new proof and detailed descriptions, especially for Riemannian metrics, highlighting their invariance properties under specific linear maps.

Contribution

It offers a new proof and comprehensive description of isometry-invariant Finsler metrics on inner product spaces, including the most general case and refined results for Riemannian metrics.

Findings

Only scalar multiples of isometries preserve the metric.

Characterization of metrics invariant under all such linear maps.

Refined results for Riemannian metrics.

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 characterise the metrics invariant with respect to all linear maps of this type.