Existence, Uniqueness and Regularity of the Projection onto Differentiable Manifolds

Abstract

We investigate the maximal open domain $\mathscr{E}(M)$ on which the orthogonal projection map $p$ onto a subset $M\subseteq \mathbb{R}^d$ can be defined and study essential properties of $p$. We prove that if $M$ is a $C^1$ submanifold of $\mathbb{R}^d$ satisfying a Lipschitz condition on the tangent spaces, then $\mathscr{E}(M)$ can be described by a lower semi-continuous 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 $k\ge 2$, the projection map is $C^{k-1}$ on $\mathscr{E}(M)$, and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order derivatives on tubular neighborhoods. A sufficient condition for the inclusion $M\subseteq\mathscr{E}(M)$ 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 $M\subseteq\mathscr{E}(M)$, 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 $\mathscr{E}(M)$ and the topological skeleton of $M^c$.