Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds

TL;DR

This paper investigates the regularity and branching behavior of abnormal geodesics in real-analytic sub-Riemannian manifolds, providing new examples and resolving open questions about their interior singularities.

Contribution

It constructs examples of geodesics with interior regularity loss and branching, and extends these findings to Carnot groups using a lifting procedure.

Findings

Existence of geodesics losing regularity at interior points.

Construction of geodesics exhibiting branching behavior.

Extension of results to Carnot groups via lifting.

Abstract

We study the regularity and branching of strictly abnormal minimizing geodesics in sub-Riemannian geometry. We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions. Moreover, using a lifting procedure, we provide the existence of non-smooth and branching minimizing geodesics also in Carnot groups.