Calculus of Variations and Geometric Measure Theory

T. Mietton - L. Rizzi

Branching geodesics in sub-Riemannian geometry

created by rizzi1 on 28 Feb 2020
modified on 02 Oct 2020


Published Paper

Inserted: 28 feb 2020
Last Updated: 2 oct 2020

Journal: Geom. Funct. Anal.
Year: 2020
Doi: 10.1007/s00039-020-00539-z

ArXiv: 2002.12293 PDF


In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure with the Heisenberg group in a suitable way. We then use this example to construct more general types of branching.