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 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 \(k\ge 2\), 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 differentials 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 \(M^c\).

我们研究了最大开域\({\mathscr {E}}(M)\)，在该域上正交投影映射p到子集\(M\subseteq {{\mathbb {R}}}^d\)可以是定义并研究p 的基本性质。我们证明，如果M是\({{\mathbb {R}}}^d\)的\(C^1\)子流形，满足切空间上的 Lipschitz 条件，则\({\mathscr {E}} (M)\)可以用一个下半连续函数来描述，称为前沿函数。我们证明如果M是\(C^2\)或者如果\(M^c\)的拓扑骨架，这个边界函数是连续的是封闭的，我们提供了一个例子，表明边界函数通常不需要连续。我们证明，对于带有\(k\ge 2\)的\(C^k\) -子流形M，投影图是\(C^{k-1}\)在\({\mathscr {E} }(M)\)，我们得到投影图的微分公式，用于讨论其高阶微分在管状邻域上的有界性。包含\(M\subseteq {\mathscr {E}}(M)\)的充分条件是M是\(C^1\)切空间满足局部 Lipschitz 条件的子流形。我们以一种新的方式证明这个条件也是必要的。更准确地说，如果M是具有\(M\subseteq {\mathscr {E}}(M)\)的拓扑子流形，则M必须是\(C^1\)并且其切线空间满足相同的局部 Lipschitz 条件。最后一部分专门强调\({\mathscr {E}}(M)\)和\(M^c\)的拓扑骨架之间的一些关系。