Pullbacks (preimages) of manifold subsets under continuous maps

[Matthias Köppe]

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.

[git]

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

[git]

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

​90f763f

ManifoldSubsetPullback.__contains__: New

[git]

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

[git]

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

​5abb86d

ManifoldSubsetPullback: More WIP

[git]

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

​e597e6d

ManifoldSubsetPullback: Allow pulling back by charts, pulling back polyehdra

[git]

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

​1623cc5

ManifoldSubsetPullback: Move computation of names to __classcall_private__

[git]

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

​dcc1f9a

ManifoldSubsetPullback: Prepare for pullbacks of opens

[git]

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

​a2dd182

ManifoldSubsetPullback: Recognize more closed sets

[git]

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

[git]

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

​3c6666a

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

[git]

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

[git]

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

​ff5eb8b

ManifoldSubsetPullback: Add doctests

[git]

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

​c268d34

ManifoldSubsetPullback: Fix example using the embedding map

[git]

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

[Matthias Köppe]

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

[git]

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

[Michael Jung]

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.

[git]

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

[git]

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

[git]

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

​6bef52b	ConvexSet_base._test_convex_set: Run the testsuite of relint
​6ab5677	RelativeInterior.is_universe: New
​c085d30	Polyhedron_base.interior: Handle the empty polyhedron correctly
​686d0af	Polyhedron_base.product: Add doctest for alias 'cartesian_product'
​2b1d108	Merge #31919
​7323b10	ConvexSet_base._test_contains: Only test extension to AA for exact base rings
​e0d2eeb	Merge #31959
​e6c3ed5	RelativeInterior.__hash__: New
​4c7d57f	src/sage/geometry/relative_interior.py: Fix doctest output
​9ea0cea	ManifoldSubsetPullback: Do not trip over relative interiors of cones

[Eric Gourgoulhon]

As revealed by the patchbot, there are some doctest failures, as well as some coverage and pyflakes issues.

[git]

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

​94e6858	RelativeInterior.ambient, ambient_vector_space, is_universe: New
​0c9bc94	ConvexSet_base: Add default implementations of ambient, ambient_dim; add doctests
​d591234	Merge #31959

[Matthias Köppe]

Thanks for taking a look! Yes, I'll have to add more documentation and examples.

[git]

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

​87f3e30

ManifoldSubsetPullback: Add documentation and tests

[git]

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

​7d3ae5c	Merge #31959
​2b1040c	ConvexSet_base.{an_element,some_elements}, Polyhedron.some_elements, RelativeInterior.some_elements: New
​a884517	PolyhedronFace.some_elements: New
​f6cc8f7	Merge #31990
​1a4805c	ManifoldSubsetPullback: For charts, use .point as the inverse
​754e002	ManifoldSubsetPullback._an_element_, some_elements: Implement

[git]

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

​a8e607c

ManifoldSubsetPullback._coord_def: Add documentation and examples

[git]

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

​ce91e44	src/sage/geometry/relative_interior.py: Fix doctest output
​b19b3cc	Merge #31959
​c49ab33	Merge #31990

[git]

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

​f0e7c58	ambient_vector_space docstring: Fix bad blocks
​200d967	ConvexSet_base.ambient doctest: Actually test the method
​1c2bef5	Merge #31959
​ed898e6	ConvexSet_base.an_element, some_elements: Go through generator method _some_elements_, make some_elements return a list
​7ae21da	Merge #31990

[Eric Gourgoulhon]

Thanks for the last changes. Some comments regarding the documentation:

• there is no entry for the class ManifoldSubsetPullback in the reference manual
• in the files continuous_map.py and subsets/pullback.py, each instance of :class:`ManifoldSubset` should be replaced by :class:`~sage.manifolds.subset.ManifoldSubset`
• the docstring of ScalarField.pullback should have an INPUT and OUTPUT sections similar to those of ContinuousMap.preimage. By the way, to be consistent with ContinuousMap.preimage, shouldn't ScalarField.pullback be renamed to ScalarField.preimage, with pullback defined as an alias to preimage?
• the INPUT and OUTPUT sections of DiffMap.pullback should be updated to take into account the new arguments.

[git]

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

​8d3ad85	Merge #31764
​60505b0	Merge #31798
​b9909c0	ManifoldSubsetClosure: Add examples, fix docstring markup
​16c6a72	ManifoldSubsetClosure, ManifoldSubset.closure: Improve documentation
​2514586	Merge #31644
​3ca139a	src/doc/en/reference/manifolds: Add pullback
​1d430ee	src/sage/manifolds/continuous_map.py, src/sage/manifolds/subsets: Use qualified class name in docstrings
​1266de9	src/sage/geometry/convex_set.py: Fix docstring markup
​df64125	ScalarField.preimage: Make pullback an alias; expand documentation
​3877c5f	ManifoldSubsetPullback.is_closed: Remove unfinished NNC_Polyhedron example

[git]

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

​0c549f0	Chart.pullback, preimage: New
​7bb720e	ManifoldSubsetPullback.__contains__: Handle the case of charts, which return tuples of coordinates, not vectors
​60ea044	ContinuousMap.preimage, ScalarField.preimage: Document that output can also be a manifold

[git]

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

​5b8ceca	InternalRealInterval, RealSet: Add _latex_ methods
​69ca854	RealSet._repr_: Use unicode cup sign instead of +
​dbdfc06	InternalRealInterval, RealSet: Remove extra whitespace in latex, add documentation
​a0e3bf5	Merge #31880
​181b128	RealSet.is_open, is_closed, closure, interior, boundary: New
​7f56338	PiecewiseFunction: Adjust doctests for changed RealSet repr
​8cd74b5	Merge #31880
​8fdb104	RealSet.boundary: Add another doctest
​9d41f0e	src/sage/sets/real_set.py: Remove unnecessary import
​c36e43b	Merge #21243

[Eric Gourgoulhon]

In the docstring of ManifoldSubsetPullback:

• in the INPUT section, there seems to be no entry for ConvexSet_base in the documentation

• in the following part of the EXAMPLES block:

  sage: r_squared = M.scalar_field(x^2+y^2)
  sage: r_squared.set_immutable()
  sage: cl_I = RealSet((1, 4)); cl_I
  (1, 4)
  sage: cl_O = ManifoldSubsetPullback(r_squared, None, I); cl_O
  

  I guess I in the last line (which, at this stage, is the imaginary number i) should be replaced by cl_I and the subsequent True and False should become False and True respectively. But then, cl_I and cl_O are identical to I and O in the example that follows...

[git]

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

​c540d35

ManifoldSubsetPullback: Fix up some doctests with closed intervals

[Matthias Köppe]

Thanks for spotting this - I have fixed this and a similar example

[Matthias Köppe]

Replying to egourgoulhon:

In the docstring of ManifoldSubsetPullback:

• in the INPUT section, there seems to be no entry for ConvexSet_base in the documentation

You may need to run ./sage -docbuild reference/discrete_geometry html

[Eric Gourgoulhon]

Replying to mkoeppe:

Replying to egourgoulhon:

In the docstring of ManifoldSubsetPullback:

• in the INPUT section, there seems to be no entry for ConvexSet_base in the documentation

You may need to run ./sage -docbuild reference/discrete_geometry html

Yes indeed!

[Eric Gourgoulhon]

Replying to mkoeppe:

Thanks for spotting this - I have fixed this and a similar example

Thanks.

At the moment, we have

sage: M = Manifold(2, 'M')                                                                          
sage: X.<x,y> = M.chart()                                                                           
sage: M.identity_map().preimage(M)                                                                  
Subset Id_M_inv_M of the 2-dimensional differentiable manifold M
sage: M.identity_map().preimage(M) is M                                                             
False
sage: M.zero_scalar_field().preimage(RealSet.point(0))                                              
Subset zero_inv_{0} of the 2-dimensional differentiable manifold M
sage: M.zero_scalar_field().preimage(RealSet.point(0)) is M                                         
False

Do you think it would be easy to have both answers to be True? If no, this should not hamper the current ticket, but be differed to another ticket.

[git]

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

​facad97

ContinuousMap.preimage: Handle identity_map specially

[Matthias Köppe]

Sure, that's easy to do and certainly a good idea. Here it is for ContinuousMap.preimage already, I'll look into the case of the scalar fields

[git]

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

​c2ecf3e

ScalarField.preimage: Handle the case of the zero scalar field

[Matthias Köppe]

More general things such as detecting that the preimage of [0,oo) under squaring is the full domain will have to wait until a follow-up ticket

[Eric Gourgoulhon]

Replying to mkoeppe:

Sure, that's easy to do and certainly a good idea. Here it is for ContinuousMap.preimage already,

Thanks. Maybe one can add

if self._is_isomorphism and codomain_subset == self._codomain:
    return self._domain

[Eric Gourgoulhon]

The patchbot reports a doctest error in src/sage/geometry/polyhedron/base.py as well as pyflakes errors in src/sage/manifolds/subsets/pullback.py.

[Matthias Köppe]

Replying to egourgoulhon:

Maybe one can add

if self._is_isomorphism and codomain_subset == self._codomain:
    return self._domain

I think I can do more generally

  if self._codomain.is_subset(codomain_subset):
      return self._domain

[Matthias Köppe]

Replying to egourgoulhon:

The patchbot reports a doctest error in src/sage/geometry/polyhedron/base.py

I've seen this one before locally, I don't think it's coming from this ticket, but I have opened #32030 for it

[git]

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

​3b1c428

ManifoldSubsetPullback: Make codomain_subset required 2nd init arg; fix pycodestyle/pyflakes warnings

[Eric Gourgoulhon]

Replying to mkoeppe:

Replying to egourgoulhon:

Maybe one can add

if self._is_isomorphism and codomain_subset == self._codomain:
    return self._domain

I think I can do more generally

  if self._codomain.is_subset(codomain_subset):
      return self._domain

Yes, indeed! Being an isomorphism is not necessary in that case.

[git]

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

​d321b93

ContinuousMap: Return domain if the map's codomain is contained in the given subset

[Eric Gourgoulhon]

A minor point: to generate correctly the documentation, the following change shall be made in the docstring of ManifoldSubsetPullback:

-    - ``map`` - an instance of :class:`ContinuousMap`,
+    - ``map`` - an instance of :class:`~sage.manifolds.continuous_map.ContinuousMap`,
       :class:`ScalarField`, or :class:`Chart`

For :class:`ScalarField` and :class:`Chart`, this is not necessary because these classes are imported in pullback.py.

[Eric Gourgoulhon]

This example in the doctstring of ManifoldSubsetPullback.is_closed is dubious:

        The pullback of finite sets is closed::

            sage: F = Family([vector(QQ, [1, 2], immutable=True), vector(QQ, [2, 3], immutable=True)])
            sage: McF = ManifoldSubsetPullback(c_cart, F, name='McF'); McF
            Subset McF of the 2-dimensional topological manifold R^2
            sage: McF.is_closed()
            False

[git]

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

​07aba9e	ManifoldSubsetPullback.is_closed: Preimage of finite sets is closed
​653c651	src/sage/manifolds/subsets/pullback.py: Fix docstring markup

[Matthias Köppe]

Replying to egourgoulhon:

This example in the doctstring of ManifoldSubsetPullback.is_closed is dubious:

Indeed. Looks like I had forgotten to actually implement this case. Done now

[Eric Gourgoulhon]

Thanks for the update. As soon as the patchbot visits the ticket and is green, the ticket can be set to positive review.

[Eric Gourgoulhon]

OK, since the changes since the last time the patchbot visited the ticket are quite minor, let us move on without waiting for another visit.

Thank you for the new features introduced here!

[Matthias Köppe]

Thanks for the review!

[Eric Gourgoulhon]

Replying to egourgoulhon:

OK, since the changes since the last time the patchbot visited the ticket are quite minor, let us move on without waiting for another visit.

It seems that we don't have any patchbot at the moment: ​https://patchbot.sagemath.org/ticket/0/

[git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

​4b930ff

Merge tag '9.4.beta4' into t/31688/pullbacks_of_manifold_subsets_under_continuous_maps

[Volker Braun]

sage -t --long --warn-long 41.5 --random-seed=0 src/sage/geometry/polyhedron/base.py
**********************************************************************
File "src/sage/geometry/polyhedron/base.py", line 2975, in sage.geometry.polyhedron.base.Polyhedron_base.incidence_matrix
Failed example:
    M == P.incidence_matrix()
Expected:
    False
Got:
    True
**********************************************************************
1 item had failures:
   1 of  31 in sage.geometry.polyhedron.base.Polyhedron_base.incidence_matrix
    [1737 tests, 1 failure, 35.30 s]
----------------------------------------------------------------------
sage -t --long --warn-long 41.5 --random-seed=0 src/sage/geometry/polyhedron/base.py  # 1 doctest failed
----------------------------------------------------------------------

[git]

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

​4558e26

Polyhedron_base.incidence_matrix: Remove accidental change to doctest