Manifolds with boundaries and corners
Most applications of differential geometry in science and engineering involve manifolds with boundary . PDEs on manifolds, for instance, are usually boundary value problems. It thus unfortunate that most books on differential geometry treat manifolds with boundary as an afterthought.
The category of smooth manifolds is bad enough but the category of smooth manifolds with boundary is worse, not evening having products. This motivates the extension to manifolds with corners. Hardly any books treat manifolds with corners. One can enlarge the category of manifolds still further in the search for a “convenient category” of smooth spaces.
Manifolds with boundary
Some authors modify basic definitions, such as those of immersions and submersions , to account for boundaries.
- Fuks, Rokhlin, 1984: Beginner’s course in topology
- A “beginner’s course” only in the Russian sense!
Manifolds with corners
The definition of a manifolds with corners is not universally agreed upon; (Joyce 2010, Remark 2.11) compares four different definitions. Neither is the definition of smooth maps betweem them (Joyce 2010, Remark 3.3).
Books
- Michor, 1980: Manifolds of differential mappings (pdf), Sec. 2: Manifolds
with corners
- Book is mostly about infinite-dimensional manifolds of mappings
- Margalef-Roig & Dominguez, 1992: Differential topology (pdf)
- Despite the unassuming title, treats differential topology on infinite-dimensional manifolds with corners
- Melrose, 1999, unfinished book: Differential analysis on manifolds with corners (website )
- Knapp, 2021: Stokes’s theory and Whitney manifolds (pdf)
- Proves Stokes’s theorem for manifolds with corners and, more generally, for Whitney manifolds
Papers
More general boundaries
Manifolds with corners, being modeled locally on the quadrant, are not as general as they may initially appear. For example, the tetrahedron is a 3-manifold with corners but the square pyramid is not.
One generalization of manifolds with corners are stratified spaces .
Another are “Whitney manifold germs”, as defined by Michor: