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

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 $\frac{n(n+1)}{2}$.

simple result; comments are welcome

References in corpus (1)

• Can we make a Finsler metric complete by a trivial projective change?