# Isometries of almost-Riemannian structures on Lie groups

## TL;DR

This paper classifies isometries of almost-Riemannian structures on Lie groups, showing that isometries fixing the identity preserve key structures, and provides a complete classification for specific groups like the 2D affine and Heisenberg groups.

## Contribution

It establishes conditions under which ARS isometries are automorphisms and offers a full classification of ARSs on certain Lie groups.

## Key findings

- Two ARSs are isometric iff an isometry fixing the identity exists.
- Isometries fixing the identity preserve the distribution and linear field.
- Complete classification of ARSs on 2D affine and Heisenberg groups.

## Abstract

A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) -- 1 left-invariant ones. It is first proven that two different ARSs are isometric if and only if there exists an isometry between them that fixes the identity. Such an isometry preserves the left-invariant distribution and the linear field. If the Lie group is nilpotent it is an automorphism. These results are used to state a complete classification of the ARSs on the 2D affine and the Heisenberg groups.