Opened 2 years ago

# Manifolds with boundary

Reported by: Owned by: mkoeppe major sage-wishlist geometry egourgoulhon, dimpase, yzh, gh-mjungmath N/A

(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:

### comment:1 Changed 2 years ago by mkoeppe

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### comment:3 Changed 16 months ago by gh-mjungmath

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

### comment:4 Changed 16 months ago by 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")

Last edited 16 months ago by mkoeppe (previous) (diff)

### comment:5 follow-ups: ↓ 6 ↓ 7 Changed 16 months ago by 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)

### comment:6 in reply to: ↑ 5 Changed 16 months ago by mkoeppe

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

Yes

### comment:7 in reply to: ↑ 5 Changed 16 months ago by mkoeppe

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

More complicated than modeling them locally by a polyhedral fan?

### comment:8 Changed 16 months ago by 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...

### comment:9 Changed 15 months ago by mkoeppe

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### comment:10 Changed 13 months ago by mkoeppe

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### comment:11 follow-up: ↓ 12 Changed 13 months ago by 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.