Almost-Riemannian Geometry on Lie Groups

Victor Ayala (1) , Philippe Jouan (2)

1 universidad catolica del Norte
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

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.

Mots clés: Almost-Riemannain geometry; Linear systems; Lie groups

Domaines: Optimisation et contrôle [math.OC]

Victor Ayala, Philippe Jouan. Almost-Riemannian Geometry on Lie Groups. SIAM Journal on Control and Optimization, 2016, ⟨10.1137/15M1038372⟩. ⟨hal-01120535v2⟩