Existence, uniqueness and regularity of the projection onto differentiable manifolds

• Published in: Annals of global analysis and geometry Vol. 60; no. 3; pp. 559 - 587
• Main Authors: Leobacher, Gunther, Steinicke, Alexander
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2021; Springer Nature B.V
• Subjects: Analysis; Applied mathematics; Continuity (mathematics); Differential Geometry; Geometry; Global Analysis and Analysis on Manifolds; Lipschitz condition; Manifolds (mathematics); Mathematical Physics; Mathematics; Mathematics and Statistics; Neighborhoods; Partial differential equations; Projection; Topology; Nonlinear orthogonal projection; Sets of positive reach; 53A07; Medial axis; 57N40; Tubular neighborhood
• ISSN: 0232-704X; 1572-9060
• DOI: 10.1007/s10455-021-09788-z

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 .