Opened 2 years ago

Manifolds with boundary

Reported by: Owned by: Matthias Köppe major sage-wishlist geometry Eric Gourgoulhon, Dima Pasechnik, Yuan Zhou, Michael Jung 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 Matthias Köppe

Cc: Dima Pasechnik Yuan Zhou added modified (diff)

comment:3 Changed 19 months ago by Michael Jung

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

comment:4 Changed 19 months ago by Matthias Köppe

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.

Version 0, edited 19 months ago by Matthias Köppe (next)

comment:5 follow-ups:  6  7 Changed 19 months ago by Dima Pasechnik

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 19 months ago by Matthias Köppe

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 19 months ago by Matthias Köppe

(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 19 months ago by Dima Pasechnik

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 18 months ago by Matthias Köppe

Description: modified (diff)

comment:10 Changed 16 months ago by Matthias Köppe

Description: modified (diff)

comment:11 follow-up:  12 Changed 16 months ago by Michael Jung

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.