Homotopic Distance and Generalized Motion Planning

Abstract We prove that the homotopic distance between two maps defined on a manifold is bounded above by sum of their subspace distances critical submanifolds any Morse–Bott function. This generalizes Lusternik–Schnirelmann theorem (for Morse functions) and similar result Farber for topological complexity. Analogously, we that, analytic manifolds, submanifold its cut locus. As an application, show how navigation functions can be used to solve generalized motion planning problem.