Opened 2 years ago
Last modified 13 months ago
#30080 new enhancement
Manifolds with boundary
Reported by: | mkoeppe | Owned by: | |
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Priority: | major | Milestone: | sage-wishlist |
Component: | geometry | Keywords: | |
Cc: | egourgoulhon, dimpase, yzh, gh-mjungmath | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
(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
Change History (12)
comment:1 Changed 2 years ago by
- Cc dimpase yzh added
- Description modified (diff)
comment:2 Changed 16 months ago by
- Cc gh-mjungmath added
comment:3 Changed 16 months ago by
comment:4 Changed 16 months ago by
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 R^{3}. 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")
comment:5 follow-ups: ↓ 6 ↓ 7 Changed 16 months ago by
hmm, what is "the top of the square pyramid in R^{3}" ?
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
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 16 months ago by
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 16 months ago by
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
- Description modified (diff)
comment:10 Changed 13 months ago by
- Description modified (diff)
comment:11 follow-up: ↓ 12 Changed 13 months ago by
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 13 months ago by
Replying to gh-mjungmath:
I think a first reasonable step would be to introduce "boundary charts".
Well, #31894 is a step into this direction
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