Subriemannian metrics and the metrizability of parabolic geometries

TL;DR

This paper explores the linearized metrizability problem within parabolic and subriemannian geometries, providing a classification method and constructing natural metrics relevant to geometric control theory.

Contribution

It introduces a general method for linearizability in parabolic geometries and classifies cases with irreducible distributions, extending classical projective geometry results.

Findings

Classification of linearizable cases with irreducible distributions

A general method for linearizability in parabolic geometries

Construction of natural subriemannian metrics for control theory

Abstract

We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville in 1889. We give a general method for linearizability and a classification of all cases with irreducible defining distribution where this method applies. These tools lead to natural subriemannian metrics on generic distributions of interest in geometric control theory.