The Myers-Steenrod theorem for Finsler manifolds of low regularity

TL;DR

This paper extends the Myers-Steenrod theorem to Finsler manifolds with minimal regularity, showing that isometries are smooth diffeomorphisms and generalizing to Finsler 1-quasiconformal maps using Binet-Legendre metrics.

Contribution

It establishes regularity results for isometries of low-regularity Finsler manifolds and generalizes the theorem to Finsler 1-quasiconformal mappings.

Findings

Isometries between low-regularity Finsler metrics are smooth diffeomorphisms.

The theorem is extended to Finsler 1-quasiconformal maps.

Reduction to Riemannian problems via Binet-Legendre metric is effective.

Abstract

We prove a version of Myers-Steenrod's theorem for Finsler manifolds under minimal regularity hypothesis. In particular we show that an isometry between $C^{k,\alpha}$-smooth (or partially smooth) Finsler metrics, with $k+\alpha>0$, $k\in \mathbb{N} \cup \{0\}$, and $0 \leq \alpha \leq 1$ is necessary a diffeomorphism of class $C^{k+1,\alpha}$. A generalisation of this result to the case of Finsler 1-quasiconformal mapping is given. The proofs are based on the reduction of the Finlserian problems to Riemannian ones with the help of the the Binet-Legendre metric.