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Differential properties of the sub-Riemannian distance function

Grochowski M.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 1

93-101

Let (M, H, g) be a sub-Riemannian manifold. Fix a point [p_0 belongs to M] and denote by f the sub-Riemannian distance from [p_0]. It is proved that f is smooth on an open and dense subset of a certain neighbourhood of a regular geodesic. On the other hand, each minimizing geodesic around which f is smooth is regular.