Existence, uniqueness and regularity of the projection onto differentiable manifolds

Gunther Leobacher, Alexander Steinicke

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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.

Original languageEnglish
Pages (from-to)559-587
Number of pages29
Journal Annals of global analysis and geometry
Issue number3
Early online date1 Jul 2021
Publication statusPublished - Oct 2021

Bibliographical note

Publisher Copyright:© 2021, The Author(s).


  • Nonlinear orthogonal projection
  • Medial axis
  • Sets of positive reach
  • Tubular neighborhood

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