Structure of the endpoint map near nice singular curves

TL;DR

This paper analyzes the local structure of the endpoint map near nice singular curves in rank-two sub-Riemannian geometry, providing a normal form and Morse index formulas to facilitate future Morse theory generalizations.

Contribution

It introduces a normal form for the endpoint map near nice singular curves and studies the Hessian's inertia index, advancing the understanding of sub-Riemannian singularities.

Findings

Normal form of the endpoint map as a sum of linear and quadratic parts

Morse-like formula for the inertia index of the Hessian

Preparation for extending Morse theory to rank-two sub-Riemannian structures

Abstract

Given a rank-two sub-Riemannian structure $(M,\Delta)$ and a point $x_0\in M$, a singular curve is a critical point of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space of horizontal curves starting at $x_0$. The typical least degenerate singular curves of these structures are called \emph{regular singular curves}; they are \emph{nice} if their endpoint is not conjugate along $\gamma$. The main goal of this paper is to show that locally around a nice singular curve $\gamma$, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which $F$ writes as a sum of a linear map and a quadratic form. We also study the restriction of $F$ to the level sets of the action functional and give a Morse-like formula for the inertia index of its Hessian at $\gamma$. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.