Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain [Formula: see text] on which the orthogonal projection map p onto a subset [Formula: see text] can be defined and study essential properties of p. We prove that if M is a [Formula: see text] submanifold of [Formula: see text] satisfying a Lipschitz condition on the tangent spaces, then [Formula: see text] can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is [Formula: see text] or if the topological skeleton of [Formula: see text] is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a [Formula: see text] -submanifold M with [Formula: see text] , the projection map is [Formula: see text] on [Formula: see text] , 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 [Formula: see text] is that M is a [Formula: see text] 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 [Formula: see text] , then M must be [Formula: see text] and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between [Formula: see text] and the topological skeleton of [Formula: see text] .