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

CC: @egourgoulhon @dimpase @yuan-zhou @mjungmath

Component: geometry

Issue created by migration from https://trac.sagemath.org/ticket/30080

[mkoeppe]

Description changed:

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+++ 
@@ -1,3 +1,7 @@
 (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").
+
+

[mjungmath]

comment:3

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]

comment:4

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]

comment:5

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]

comment:6

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]

comment:7

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]

comment:8

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...

[mkoeppe]

Description changed:

--- 
+++ 
@@ -5,3 +5,9 @@
 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
+

[mkoeppe]

Description changed:

--- 
+++ 
@@ -10,4 +10,4 @@
 - 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

[mjungmath]

comment:11

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]

comment:12

Replying to @mjungmath:

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

Well, #31894 is a step into this direction