Almost-Riemannian Geometry on Lie Groups

TL;DR

This paper studies Almost-Riemannian Structures on Lie groups, analyzing their singularities, extremals, and desingularization, with extensions to homogeneous spaces and illustrative examples on specific groups.

Contribution

It introduces a simple ARS on Lie groups, characterizes singularities and extremals, and extends the framework to homogeneous spaces with necessary and sufficient conditions.

Findings

Characterization of the singular locus and abnormal extremals.

Desingularization methods for ARS on Lie groups.

Extension of ARS to homogeneous spaces with criteria for equivalence.

Abstract

A simple Almost-Riemmanian Structure on a Lie group G is defined by a linear vector field and dim(G)-1 left-invariant ones. We state results about the singular locus, the abnormal extremals and the desingularization of such ARS's, and these results are illustrated by examples on the 2D affine and the Heisenberg groups.These ARS's are extended in two ways to homogeneous spaces, and a necessary and sufficient condition for an ARS on a manifold to be equivalent to a general ARS on a homogeneous space is stated.