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⊆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 C^k-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⊆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⊆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 .