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

Algebraic Topology