Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain E(M) on which the orthogonal projection map p onto a subset M subset of R-d can be defined and study essential properties of p. We prove that if M is a C-1 submanifold of R-d satisfying a Lipschitz condition on the tangent spaces, then E(M) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C-2 or if the topological skeleton of M-c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C-k-submanifold M with k >= 2, the projection map is Ck-1 on E(M), and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion M subset of E(M) is that M is a C-1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with M subset of E(M), then M must be C-1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between E(M) and the topological skeleton of M-c.