Abstract
In this paper we study a class of sub-spaces 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 Hδrmander'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 Carnot-Caratheodory manifold is sub-analytic; (2) a study of the singularity of the horizontal loop space. At the end we study the geometry of left-invarian t Carnot-Caratheodory metrics on fact nilpotent groups.
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