Normal Singular Geodesics of a Conformally Generic Sub-Riemannian Metric

TL;DR

This paper proves that under generic conditions, certain real-analytic sub-Riemannian metrics do not admit non-trivial normal singular geodesics, especially when the distribution has co-rank one, impacting the understanding of geodesic behavior.

Contribution

It establishes the absence of non-trivial normal singular geodesics for generic real-analytic Hamiltonians with totally non-holonomic distributions, extending the understanding of geodesic regularity.

Findings

No non-trivial normal D-singular orbits of minimal rank for generic Hamiltonians.

In co-rank 1 cases, adding a generic potential eliminates non-trivial normal D-singular orbits.

Results apply to real-analytic, totally non-holonomic distributions, influencing sub-Riemannian geometry.

Abstract

We prove that a Ma\~n\'e generic real-analytic $D$-Hamiltonian H, subjected to a totally non-holonomic real-analytic distribution $D$, has no non-trivial normal $D$-singular orbits of minimal rank. If $D$ has co-rank 1, this implies that $H + u$, where $u$ is a generic real-analytic potential, does not admit non-trivial normal $D$-singular orbits.