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.

中文翻译：

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次黎曼几何中的分支测地线

在此注释中，我们表明子黎曼流形可以包含分支法线最小化测地线。当且仅当正常测地线的等级在非零时间不连续时才会发生此现象，特别是对于严格的正常测地线意味着它包含一个非平凡的异常子段。最简单的示例是通过以合适的方式将三维Martinet平面结构与Heisenberg组粘合在一起获得的。然后，我们使用此示例构造更一般的分支类型。