On submaximal dimension of the group of almost isometries of Finsler metrics

TL;DR

This paper determines the maximum possible dimensions of groups of almost isometries for Finsler metrics, showing that higher symmetry implies the metric is Randers with specific curvature properties.

Contribution

It establishes the second greatest dimension of almost isometry groups for Finsler metrics and characterizes metrics with larger symmetry as Randers with constant curvature.

Findings

Maximum dimension of almost isometry group is (n^2 - n)/2 + 1 for n ≠ 4

Metrics with larger symmetry are Randers with constant sectional curvature

Special case for n=4 where the maximum dimension is 8

Abstract

We show that the second greatest possible dimension of the group of (local) almost isometries of a Finsler metric is $\frac{n^2 -n}{2} +1$ for $n= dim(M)\ne 4 $ and $\frac{n^2 -n}{2} +2 =8$ for $n=4$. If a Finsler metric has the group of almost isometries of dimension greater than $\frac{n^2 -n}{2} +1$, then the Finsler metric is Randers, i.e., $F(x,y)= \sqrt{g_x(y,y)} + \tau(y)$. Moreover, if $n\ne 4$, the Riemannian metric $g$ has constant sectional curvature and, if in addition $n\ne 2$, the 1-form $\tau$ is closed, so (locally) the metric admits the group of local isometries of the maximal dimension $\frac{n(n+1)}2$.