General Nilpotent and Solvable Approximations of Almost-Riemannian Structures

TL;DR

This paper investigates how almost-Riemannian structures can be approximated by nilpotent or solvable models, revealing their linear nature on Lie groups or homogeneous spaces and analyzing generic properties across dimensions.

Contribution

It demonstrates that nilpotent and solvable approximations of almost-Riemannian structures are always linear on Lie groups or homogeneous spaces and identifies their generic properties in all dimensions.

Findings

Nilpotent or solvable approximations are linear on Lie groups or homogeneous spaces.

Generic properties of almost-Riemannian structures are characterized across all dimensions.

Identification of generic nilpotent and solvable approximations.

Abstract

It is first shown that the nilpotent or the solvable approximation of an almost-Riemannian structure at a singular point is always a linear almost-Riemannian structure on a Lie group or a homogeneous space. The generic properties of almost-Riemannian structures are then investigated in all dimensions and the generic nilpotent and solvable approximations are identified.