Pullbacks (preimages) of manifold subsets under continuous maps

[mkoeppe]

Similar to #31653, given a continuous map \Phi: N -> M and a manifold subset S of M, we define the pullback (preimage) of S as the subset of N of points p with \Phi(p) in S.

Given a real scalar field Phi: N -> R and a RealSet S, we define the pullback in the same way.

Also, we view a chart C as a continuous function Phi: C.domain() -> R^n and allow pulling back any subset of R^n (an object with a __contains__ method; for example polyhedra, lattices, linear subspaces, tensor modules) by it as well.

In all cases, because Phi is continuous, topological closures/interiors pull back.

An application is in #31981.

Depends on #31883 Depends on #31904 Depends on #31653 Depends on #31916 Depends on #31644 Depends on #31959 Depends on #31990 Depends on #21243

CC: @egourgoulhon @tscrim @mjungmath

Component: manifolds

Author: Matthias Koeppe

Branch/Commit: 4558e26

Reviewer: Eric Gourgoulhon

Issue created by migration from https://trac.sagemath.org/ticket/31688

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,2 +1,7 @@
 Similar to #31653, given a continuous map `\Phi: N -> M` and a manifold subset `S` of `M`, we define the pullback (preimage) of `S` as the subset of `N` of points `p` with `\Phi(p)` in `S`.

+Given a real scalar field `Phi: N -> R` and a RealSet `S`, we define the pullback in the same way.
+
+In both cases, because `Phi` is continuous, topological closures/interiors pull back.
+
+

[mkoeppe]

Dependencies: #31883

[mkoeppe]

Branch: u/mkoeppe/pullbacks_of_manifold_subsets_under_continuous_maps

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Commit: 941001b

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

1942c71	ScalarField.codomain: New, put scalar fields in category of continuous maps
8829d72	Merge #31883
941001b	ManifoldSubsetPullback: More WIP

[mkoeppe]

Author: Matthias Koeppe

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,6 +1,6 @@
 Similar to #31653, given a continuous map `\Phi: N -> M` and a manifold subset `S` of `M`, we define the pullback (preimage) of `S` as the subset of `N` of points `p` with `\Phi(p)` in `S`.

-Given a real scalar field `Phi: N -> R` and a RealSet `S`, we define the pullback in the same way.
+Given a real scalar field `Phi: N -> R` and a `RealSet` `S`, we define the pullback in the same way.

 In both cases, because `Phi` is continuous, topological closures/interiors pull back.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 941001b to 90f763f

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

90f763f

ManifoldSubsetPullback.__contains__: New

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

ff34765	ManifoldSubsetPullback: In `__classcall_private__`, make immutable copies of the maps
b8dadf3	ScalarField.pullback, ContinuousMap.pullback: New

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 90f763f to b8dadf3

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

5abb86d

ManifoldSubsetPullback: More WIP

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from b8dadf3 to 5abb86d

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

e597e6d

ManifoldSubsetPullback: Allow pulling back by charts, pulling back polyehdra

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 5abb86d to e597e6d

[mkoeppe]

Description changed:

--- 
+++ 
@@ -2,6 +2,8 @@

 Given a real scalar field `Phi: N -> R` and a `RealSet` `S`, we define the pullback in the same way.

-In both cases, because `Phi` is continuous, topological closures/interiors pull back.
+Also, we view a chart `C` as a continuous function `Phi: C.domain() -> R^n` and allow pulling back by it as well. 
+
+In all cases, because `Phi` is continuous, topological closures/interiors pull back.

[mkoeppe]

Description changed:

--- 
+++ 
@@ -2,7 +2,7 @@

 Given a real scalar field `Phi: N -> R` and a `RealSet` `S`, we define the pullback in the same way.

-Also, we view a chart `C` as a continuous function `Phi: C.domain() -> R^n` and allow pulling back by it as well. 
+Also, we view a chart `C` as a continuous function `Phi: C.domain() -> R^n` and allow pulling back any subset of `R^n` (an object with a `__contains__` method; for example polyhedra) by it as well. 

 In all cases, because `Phi` is continuous, topological closures/interiors pull back.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

1623cc5

ManifoldSubsetPullback: Move computation of names to __classcall_private__

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from e597e6d to 1623cc5

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

dcc1f9a

ManifoldSubsetPullback: Prepare for pullbacks of opens

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 1623cc5 to dcc1f9a

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from dcc1f9a to a2dd182

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

a2dd182

ManifoldSubsetPullback: Recognize more closed sets

[mkoeppe]

Changed dependencies from #31883 to #31883, #31904

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

459758e	ImageManifoldSubset, ContinuousMap.image: Add optional 'inverse' argument, use it in __contains__
26c7e56	src/sage/manifolds/continuous_map_image.py: Update doctests
3e273bb	TopologicalSubmanifold.as_subset: New
9726d36	Docstring work
19762ae	ImageManifoldSubset: New parameter domain_subset, use it in ContinuousMap.image
964f9f7	src/sage/manifolds/continuous_map.py: Update copyright
e711215	src/sage/manifolds/continuous_map_image.py: Add tests
0f3e36d	Link in documentation of sage.manifolds.continuous_map_image
4a13d8b	Merge #31653
79232db	ContinuousMap.preimage: New, make pullback an alias; add example

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from a2dd182 to 79232db

[mkoeppe]

Changed dependencies from #31883, #31904 to #31883, #31904, #31653

[mkoeppe]

Description changed:

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

3c6666a

ManifoldSubsetPullback: Make pullback of open RealSet under a ScalarField an open set

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 79232db to 3c6666a

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Changed commit from 3c6666a to e95c897

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

235d75a	ManifoldSubsetPullback.closure: Adjust doctest
aea4554	#31904: Fix indentation in _pullback_chart
e95c897	Merge #31904

[mkoeppe]

Description changed:

--- 
+++ 
@@ -2,7 +2,7 @@

 Given a real scalar field `Phi: N -> R` and a `RealSet` `S`, we define the pullback in the same way.

-Also, we view a chart `C` as a continuous function `Phi: C.domain() -> R^n` and allow pulling back any subset of `R^n` (an object with a `__contains__` method; for example polyhedra) by it as well. 
+Also, we view a chart `C` as a continuous function `Phi: C.domain() -> R^n` and allow pulling back any subset of `R^n` (an object with a `__contains__` method; for example polyhedra, lattices, or linear subspaces) by it as well. 

 In all cases, because `Phi` is continuous, topological closures/interiors pull back.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from e95c897 to ff5eb8b

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

ff5eb8b

ManifoldSubsetPullback: Add doctests

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from ff5eb8b to c268d34

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

c268d34

ManifoldSubsetPullback: Fix example using the embedding map

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

56b22a3	ManifoldSubsetPullback.__classcall_private__: Add doc
762e0a8	ManifoldSubsetPullback.__init__: Add tests
2a01a07	ManifoldSubsetPullback._an_element_: New

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Changed commit from c268d34 to 2a01a07

[mkoeppe]

comment:24

This is not quite ready yet, but comments are very welcome

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 2a01a07 to acc71c3

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

e9c670c	sage.geometry.polyhedron.relint, Polyhedron_base.relative_interior, Polyhedron_base.interior: New
86fc6cd	Merge #31916
acc71c3	ManifoldSubsetPullback: Pulling back interiors of polyhedra

[mkoeppe]

Changed dependencies from #31883, #31904, #31653 to #31883, #31904, #31653, #31916

[mjungmath]

comment:27

For a possible follow-up, it might be a good idea to bear in mind that the preimage of a regular value of a differentiable map between manifolds of dimension n and m respectively is a differentiable manifold of dimension n-m again.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

9df2104	sage.geometry.relative_interior: Move here from sage.geometry.polyhedron.relint
b8bfe20	ConvexRationalPolyhedralCone: Add methods interior, relative_interior
6869673	relative_interior: Fix for dimension 0
021d073	RelativeInterior: Add documentation, tests, comparison methods, method relative_interior
8f38e04	ConvexRationalPolyhedralCone.interior, relative_interior: Add doctests
3a5870e	Merge #31916

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Changed commit from acc71c3 to 3a5870e

[mkoeppe]

Description changed:

--- 
+++ 
@@ -2,7 +2,7 @@

 Given a real scalar field `Phi: N -> R` and a `RealSet` `S`, we define the pullback in the same way.

-Also, we view a chart `C` as a continuous function `Phi: C.domain() -> R^n` and allow pulling back any subset of `R^n` (an object with a `__contains__` method; for example polyhedra, lattices, or linear subspaces) by it as well. 
+Also, we view a chart `C` as a continuous function `Phi: C.domain() -> R^n` and allow pulling back any subset of `R^n` (an object with a `__contains__` method; for example polyhedra, lattices, linear subspaces, tensor modules) by it as well. 

 In all cases, because `Phi` is continuous, topological closures/interiors pull back.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

d11280e	src/sage/manifolds/subsets/pullback.py: Fixup import
cd3ca79	Update doctests for refined category of ScalarField
105bb8b	Merge #31883
5f9c852	RelativeInterior.interior: New
5c089ec	RelativeInterior.__eq__, __ne__: Handle comparisons with objects of other types
86ce301	Polyhedron_base.is_relatively_open: New; fix relative_interior for affine subspaces
216cb81	ConvexRationalPolyhedralCone.is_relatively_open: New, fix ConvexRationalPolyhedralCone.relative_interior for linear subspaces
44cde1e	src/doc/en/reference/discrete_geometry/index.rst: Add sage/geometry/relative_interior
fa4c2d2	Whitespace fixes
e802a21	Merge #31916

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 3a5870e to e802a21

[mkoeppe]

Description changed:

--- 
+++ 
@@ -7,3 +7,5 @@
 In all cases, because `Phi` is continuous, topological closures/interiors pull back.

+An application is in #31981.
+