Opened 16 months ago

Last modified 3 months ago

#30080 new enhancement

Manifolds with boundary

Reported by: mkoeppe Owned by:
Priority: major Milestone: sage-wishlist
Component: geometry Keywords:
Cc: egourgoulhon, dimpase, yzh, gh-mjungmath Merged in:
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Report Upstream: N/A Work issues:
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Description (last modified by 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:

Change History (12)

comment:1 Changed 16 months ago by mkoeppe

  • Cc dimpase yzh added
  • Description modified (diff)

comment:2 Changed 6 months ago by mkoeppe

  • Cc gh-mjungmath added

comment:3 Changed 6 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.

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

comment:4 Changed 6 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 6 months ago by mkoeppe (previous) (diff)

comment:5 follow-ups: Changed 6 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 6 months ago by 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

comment:7 in reply to: ↑ 5 Changed 6 months ago by 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?

comment:8 Changed 6 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 5 months ago by mkoeppe

  • Description modified (diff)

comment:10 Changed 3 months ago by mkoeppe

  • Description modified (diff)

comment:11 follow-up: Changed 3 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.

comment:12 in reply to: ↑ 11 Changed 3 months ago by 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

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