Differentiable manifolds: basics

[egourgoulhon]

This is the first ticket about the implementation of differentiable manifolds resulting from the SageManifolds project. See the metaticket #18528 for an overview.

The base field K of the differentiable manifold is generic (only assumed to be some non-discrete topological field), so that the user may specify e.g. K=R (real manifolds) or K=C (complex manifolds). This ticket implements the following Python classes, all of them being subclasses of classes introduced for topological manifolds (tickets #18529, #18640, #18725):

• DifferentiableManifold (subclass of TopologicalManifold, cf. Topological manifolds: basics #18529): differentiable
  manifold over a topological field K (Parent class)
• DiffChart (subclass of Chart, cf. Topological manifolds: basics #18529): chart of a K-differentiable atlas
  • RealDiffChart (subclass of RealChart, cf. Topological manifolds: basics #18529): chart of a K-differentiable atlas
    for K=R
• DiffCoordChange (subclass of CoordChange, cf. Topological manifolds: basics #18529): differentiable transition map
• DiffScalarFieldAlgebra (subclass of ScalarFieldAlgebra, cf. Topological manifolds: scalar fields #18640): set C^k(M) of
  k-times continuously K-differentiable functions M --> K, where M is a differentiable manifold
  over K, C^k(M) being a commutative algebra over K (Parent class)
• DiffScalarField (subclass of ScalarField, cf. Topological manifolds: scalar fields #18640): k-times continuously
  K-differentiable function M --> K (Element class)
• DiffManifoldHomset (subclass of TopManifoldHomset, cf. Topological manifolds: morphisms #18725): set Hom(M,N) of
  differentiable maps between the differentiable manifolds M and N over the same topological
  field K (Parent class)
• DiffMap (subclass of ContinuousMap, cf. Topological manifolds: morphisms #18725): differentiable map M --> N (Element class)

The follow-up ticket is #18843.

Documentation:
The reference manual is produced by
sage -docbuild reference/manifolds html
It can also be accessed online at http://sagemanifolds.obspm.fr/doc/18783/reference/manifolds/
More documentation (e.g. example worksheets) can be found here.

Depends on #18725
Depends on #18175

CC: @sagetrac-mbejger @sagetrac-bpillet

Component: geometry

Keywords: differentiable manifolds

Author: Eric Gourgoulhon, Michal Bejger

Branch/Commit: 97172dd

Reviewer: Volker Braun

Issue created by migration from https://trac.sagemath.org/ticket/18783

[sagetrac-git]

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

f4c9f0f

Start to improve the documentation of differentiable manifolds

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

999cac5

Minor modifications in CoordFunctionSymb and ContinuousMap.__invert__

[sagetrac-git]

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

8c8bfcd

Improve documentation of differentiable manifolds

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127b1e0

Plot of coordinate charts and points

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f2fef7b

Small improvements in the documentation of differentiable manifolds

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8ab80d8

Improvement in simplify_sqrt_real(); minor modif. in documentation

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

2f231b6

Major improvements in the documentation of diff. manifolds (basic part)

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

30e812a	Minor modifications in CoordFunctionSymb and ContinuousMap.__invert__
c0b3284	Improve documentation of classes Chart and ContinuousMap
4c72118	Add method plot() to RealChart and TopManifoldPoint
055bf3e	Improve documentation of RealChart.plot() and TopManifoldPoint.plot()
57b21df	Add doctest to set_axes_labels
e558c06	Improvement in simplify_sqrt_real(); TestSuite run non-verbose
d65f654	Introduce open covers on top manifolds + many improvements in the documentation
00c327d	Slight reorganization of the reference manual of topological manifolds (morphisms part)
f8d3f27	Merge #18725 into #18640
f0ca2de	Merge #18783 into #18725

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

8b851a0	Merge branch 'develop' into public/categories/topological_metric_spaces-18175
f8f5b93	Fixing last remaining doctests.
041a5d1	Adding p-adics to metric spaces and some cleanup.
bfa0cdf	One last doc tweak.
d13c368	Fixing doc of metric spaces.
2605c0b	Merge #18529 (Topological manifolds: basics) into #18175 (Implement categories for topological...)
6dec6d5	Implement topological manifolds (basics, #18529) on the new categories for manifolds (#18175)
3403978	Implement top. manifolds (scalar fields, #18640) on the new manifold categories (#18175)
b0521ef	Implement top. manifolds (morphisms, #18725) on the new manifold categories (#18175)
f643097	Implement diff. manifolds (basics, #18783) on the new manifold categories (#18175)

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

902908b	Revert to UniqueRepresentation for topological manifolds and charts, with the possibility to reuse manifold names.
252e616	Remove UniqueRepresentation, leaving only WithEqualityById, for topological manifolds and charts.
e7139ab	Merge branch top_manif_basics without UniqueRepresentation into top_manif_scalar_fields.
e2f192f	Remove UniqueRepresentation, leaving only WithEqualityById, for scalar fields on topological manifolds
668bc26	Remove UniqueRepresentation, leaving only WithEqualityById, for topological manifold homsets
6518699	Introduce the attribute _field_type in class TopologicalManifold to check for real and complex manifolds.
22383e6	Check for real/complex manifold performed on base_field_type() instead of RR/CC
66f2c5a	Change function('f', x) to function('f')(x) to accomodate the deprecation warning introduced in #17447
a28ed04	Morphisms of topological manifolds with the use of base_field_type()
f31bed1	Remove UniqueRepresentation from differentiable manifolds

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

85d03dc	Change the argument type to structure in Manifold
5251ef0	Remove method _test_pickling from class TopologicalManifoldPoint
f69c9ee	Fix doctest error in coord_func_symb.py due to #19312 (update to pynac-0.5.2)
7889a5d	Change in simplify_sqrt_real to cope with the change of != operator induced by #19312 (Sage 6.10.beta7)
e8d2ba6	Differentiable manifolds: basics with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)

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

0fb39df	Refactoring the code to separate out the structural part of the manifold.
cdfa817	Merge branch u/tscrim/top_manifolds_refactor into Sage 6.10.rc1 + public/manifolds/top_manif_basics
c5f35af	Make AbstractSet inherit from UniqueRepresentation; correct doctests; start to change documentation.
3a52500	Some class renaming; add more examples and doctests (full coverage)
a1ba52a	Replace a!=b by not(a==b) in simplify_sqrt_real to cope with the change of logic induced by #19312 (Sage 6.10.beta7)
e2c7ab3	Scalar fields on the refactored topoological manifolds
cb53417	Add argument full_name to AbstractNamedObject; remove _repr_() from all parent classes; improve documentation
ce03503	Scalar fields on topological manifolds: slight improvements in the documentation
bc77059	Morphisms on the refactored topoological manifolds
2df7af7	Refactorization of differentiable manifolds, with the mixin class DifferentiableMixin

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

c38ae80	Suppress the (unused) argument category in OpenTopologicalSubmanifold; minor doc improvements
069baf4	Merge branch 'public/manifolds/top_manif_basics' into Sage 7.0.beta2.
19caedb	Revert to AbstractNamedObject without argument full_name; restore _repr_() in manifold classes
9d25aa0	Scalar field algebras with AbstractNamedObject without argument full_name
ed4aa58	ScalarFieldAlgebra does not longer inherit from AbstractNamedObject
f6c6a34	Topological manifold morphisms with AbstractNamedObject without full_name
7204572	Basics of diff. manifolds with AbstractNamedObject without full_name

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

3cd03a4	Add methods lift() and retract() to ManifoldSubset; add method __eq__() in CoordChange
984c3c2	Revert to simple hierarchy for manifold classes
c01048f	Scalar fields with the simplified hierarchy for manifold classes
c866d6c	Morphisms of topological manifolds with the simplified hierarchy for manifold classes
0e04631	Basics of diff. manifolds with the simplified hierarchy for manifold classes

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

8e17d54	Merge into the latest version of #18529; improve treatment of composite functions in ExpressionNice
f00be00	Topological manifold morphisms: solved merge conflict with Sage 7.1.beta1
21b3968	Basics of diff. manifolds: solve merge conflict with Sage 7.1.beta1

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

33abe9e	Marking known bugs in utilities.py.
57b7b7b	Implementing global_options for display formatting.
f11df50	Set CoordFunctionSymbRing in the category of commutative algebras over SR; minor documentation changes
f3f5470	Fixing a (essentially trivial) doctest failure.
7b3dab3	Merge branch 'public/manifolds/top_manif_scalar_fields' of git://trac.sagemath.org/sage into Sage 7.2.rc0
9ec7d3e	Python 3 format for print in manifolds
d190b37	Morphisms of topological manifolds with coordinate functions as algebra elements
2577fc5	Fixing a (essentially trivial) doctest failure.
7125e29	Python3 format for print in morphims of topological manifolds
bd2f35a	Diff. manifolds with coordinate functions as algebra elements

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

4fad094	Merge branch 'public/manifolds/top_manif_morphisms' of trac.sagemath.org:sage into public/manifolds/top_manif_morphisms
cfecb18	Reviewer changes and tweaks for continuous maps ticket.
84051b0	Merge branch 'public/manifolds/diff_manif_basics' of trac.sagemath.org:sage into public/manifolds/diff_manif_basics

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

efcb618	Modify authorship for continuous maps.
97172dd	Basics of differentiable manifolds with changes in morphisms of topological manifolds

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[vbraun]

Reviewer: Volker Braun

[vbraun]

Changed branch from public/manifolds/diff_manif_basics to 97172dd