Branching Geodesics in Sub-Riemannian Geometry
Copyright policy )Branching Geodesics in Sub-Riemannian Geometry
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.
11 pages, 1 figure. Final version, to appear on Geometric And Functional Analysis (GAFA)
Controllability, Mathematics - Differential Geometry, Transformations, normal geodesics, 49J15, Metric Geometry (math.MG), Optimality conditions for problems involving ordinary differential equations, 510, Sub-Riemannian geometry, 53C17, Existence theories for optimal control problems involving ordinary differential equations, Mathematics - Metric Geometry, Differential Geometry (math.DG), 53C17, 49J15, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Attainable sets, reachability, FOS: Mathematics, sub-Riemannian manifold, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], branching geodesics
Controllability, Mathematics - Differential Geometry, Transformations, normal geodesics, 49J15, Metric Geometry (math.MG), Optimality conditions for problems involving ordinary differential equations, 510, Sub-Riemannian geometry, 53C17, Existence theories for optimal control problems involving ordinary differential equations, Mathematics - Metric Geometry, Differential Geometry (math.DG), 53C17, 49J15, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Attainable sets, reachability, FOS: Mathematics, sub-Riemannian manifold, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], branching geodesics
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).5 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 10% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!