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Horizontal path spaces and Carnot-Carathéodory metrics. (English) Zbl 0797.49033
We study a class of subspaces of loop spaces which have appeared in the calculus of variations. Generalizing a result of Smale, we show that the space of loops tangent to a distribution satisfying Hörmander’s condition is weakly homotopic to the space of all loops. If the distribution is fat, we resolve the end point map from the space of horizontal paths. This resolution has two applications: (1) the proof that the cut-locus on an analytic fat Carno-Carathéodory manifold is subanalytic; (2) a study of the singularity of the horizontal loop space. At the end we study the geometry of left-invariant Carnot-Carathéodory metrics on fat nilpotent groups.
Reviewer: Zhong Ge

MSC:
49N45 Inverse problems in optimal control
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