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Branching Geodesics in Sub-Riemannian Geometry
Geometric and Functional Analysis  (IF2.148),  Pub Date : 2020-08-09, DOI: 10.1007/s00039-020-00539-z
Thomas Mietton, Luca Rizzi

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.