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 .