Manifolds with boundary

[mkoeppe]

(from #30061)

We propose to add manifolds with boundary to sage.manifolds.

Simple examples of topological manifolds with boundary include convex polyhedra and semialgebraic sets with non-singular boundary. These are (except in special cases) not differentiable manifolds, but only "piecewise differentiable" ("manifolds with corners").

References:

• Dominic Joyce, On manifolds with corners, ​https://arxiv.org/pdf/0910.3518.pdf, 34 pages
• ​http://www-math.mit.edu/~rbm/18.158/daomwc.1/daomwc.1.pdf, 38 pages
• Dominic Joyce, A generalization of manifolds with corners, Advances in Mathematics, Volume 299, 20 August 2016, Pages 760-862, ​https://www.sciencedirect.com/science/article/pii/S0001870816307186
  • a useful reference for the connection to toric varieties that appear as model spaces - ​https://dacox.people.amherst.edu/lectures/coxcimpa.pdf

[gh-mjungmath]

Perhaps it is better to implement manifolds with corners right away since manifolds with boundaries are just a special case.

​https://ncatlab.org/nlab/show/manifold+with+boundary

[mkoeppe]

​https://arxiv.org/pdf/0910.3518.pdf (Remark 2.11) has a nice overview over several inequivalent definitions of manifolds with corners.

I haven't checked the details yet but I would be interested in a definition that generalizes all polyhedra, including those with degenerate vertices such as the top of the square pyramid in R³. The main definition in this paper, 2.1(iii), does not fit the bill; it would only include simple polyhedra.

A newer article by the same author: ​https://www.sciencedirect.com/science/article/pii/S0001870816307186 on manifolds with "generalized corners" ("g-corners")

[dimpase]

hmm, what is "the top of the square pyramid in R³" ?

Do you mean to say that you'd like a definition that includes all the non-simple polytopes, at least?

(vertices of non-convex polyhedra are a different story, much more complicated)

[mkoeppe]

Replying to dimpase:

Do you mean to say that you'd like a definition that includes all the non-simple polytopes, at least?

Yes

[mkoeppe]

Replying to dimpase:

(vertices of non-convex polyhedra are a different story, much more complicated)

More complicated than modeling them locally by a polyhedral fan?

[dimpase]

should one call a vertex the point in the middle of the twised prism one gets from enough twisting? If you do, you get a vertex in the middle of an edge. If you don't, you get facets without an orientation...

[gh-mjungmath]

I think a first reasonable step would be to introduce "boundary charts".

Tbh, I don't know how sensible it is to start with the most general concept of "boundary-like". Manifolds with corners seem fairly doable. The generalization by Joyce looks very interesting, though I reckon it's pretty hard to implement.

[mkoeppe]

Replying to gh-mjungmath:

I think a first reasonable step would be to introduce "boundary charts".

Well, #31894 is a step into this direction