Branching Geodesics in Sub-Riemannian Geometry
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GAFA Geometric And Functional Analysis
BRANCHING GEODESICS IN SUB-RIEMANNIAN GEOMETRY Thomas Mietton and Luca Rizzi
Abstract. 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.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1 Branching and magnetic fields. . . . . . . . . . 1.2 Strictly abnormal branching. . . . . . . . . . . 2 Sub-Riemannian Geometry . . . . . . . . . . . . . . 2.1 Characterization of geodesics. . . . . . . . . . . 3 Branching Geodesics . . . . . . . . . . . . . . . . . . 4 An Example of Branching Strictly Normal Geodesic 5 Normal Geodesics with Multiple Branching . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction A metric space is branching if there exist minimal geodesics starting from the same point which follow the same path for some initial time interval, and then become disjoint. Common examples are found in Finsler geometry (e.g. R2 with the sup norm), or on graphs. On the other hand, it is well-known that Riemannian manifolds and Alexandrov spaces with curvature bounded from below are non-branching. We are interested here in sub-Riemannian spaces, a large class of metric structures generalizing Riemannian geometry where a metric is defined only on a subset of tangent directions (cf. Sect. 2 for precise definitions). Several questions concerning geodesics, which are trivial in Riemannian geometry, become hard open problems in Mathematics Subject Classification: 53C17, 49J15
T. MIETTON, L. RIZZI
GAFA
the sub-Riemannian setting. For example it has been only recently proven in [HLD16] that sub-Riemannian geodesics cannot have corners, but it is not yet known whether geodesics are C 1 , see for example [Rif17]. To provide further motivation for our contribution, let us mention that there is an on-going effort in trying to define a suitable concept of lower curvature bound for subRiemannian spaces, in particular in relation with the synthetic approach a` la Lott– Sturm–Villani (cf. for example [BR19, BKS18, BKS19, BG17, Mil19] and [Vil19, p. 1127–1143]). Since the existence of branching geodesics causes difficulties in the study of optimal transport and of spaces satisfying synthetic curvature bounds, it is important for further progress in the theory to understand whethe
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