Existence, uniqueness and regularity of the projection onto differentiable manifolds

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