Differentiable manifolds: tangent spaces

[egourgoulhon]

This ticket implements tangent vectors and tangent spaces on differentiable manifolds. This is a follow-up of #18843 within the SageManifolds project (see the metaticket #18528 for an overview). As in #18843, the non-discrete topological field K over which the differentiable manifold is defined is generic, although in most applications, K=R or K=C.

This ticket implements the following Python classes:

• the parent class TangentSpace for tangent spaces to a differentiable manifold
• the element class TangentVector for tangent vectors to a differentiable manifold
  It also implements
• the method at() in tensor field classes (introduced in Differentiable manifolds: vector fields and tensor fields #18843) to take the value of a given
  tensor field (in particular a vector field) at a given point; the method at() is
  implemented for vector frames and coframes as well.
• the method plot() in class VectorField, to get some graphical representations of vector
  fields
• the methods differential(), differential_functions() and jacobian_matrix() to the class
  DiffMap introduced in Differentiable manifolds: basics #18783 (differential of a map between differentiable manifolds at a
  given point).

Documentation:
The reference manual is produced by
sage -docbuild reference/manifolds html
It can also be accessed online at http://sagemanifolds.obspm.fr/doc/19092/reference/manifolds/
More documentation (e.g. example worksheets) can be found here.

Depends on #15916
Depends on #18843

CC: @sagetrac-mbejger

Component: geometry

Keywords: differentiable manifold, tangent vector, tangent space

Author: Eric Gourgoulhon, Michal Bejger, Marco Mancini

Branch/Commit: ff6eb09

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/19092

[sagetrac-git]

Changed commit from d5562a2 to 6a1fc97

[sagetrac-git]

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

6a1fc97

Add method differential() to class DiffMap

[sagetrac-git]

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

31d6789

Add method plot() to class VectorField; full doctest coverage.

[sagetrac-git]

Changed commit from 6a1fc97 to 31d6789

[egourgoulhon]

Changed dependencies from #15916, #18100, #18843 to #15916, #18843

[sagetrac-git]

Changed commit from 31d6789 to 04360b0

[sagetrac-git]

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

04360b0

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

[sagetrac-git]

Changed commit from 04360b0 to 513f5ba

[sagetrac-git]

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

513f5ba

Improve TensorField.__eq__ (case with no open cover known)

[sagetrac-git]

Changed commit from 513f5ba to e43e03b

[sagetrac-git]

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

00c327d	Slight reorganization of the reference manual of topological manifolds (morphisms part)
f8d3f27	Merge #18725 into #18640
f2fef7b	Small improvements in the documentation of differentiable manifolds
8ab80d8	Improvement in simplify_sqrt_real(); minor modif. in documentation
2f231b6	Major improvements in the documentation of diff. manifolds (basic part)
f0ca2de	Merge #18783 into #18725
307e655	Major improvements in the documentation of diff. manifolds (tensor field part)
dc7f7a1	Improve TensorField.__eq__ (case with no open cover known)
4d6f21c	Merge #18843 into #18783
e43e03b	Merge #19092 into #18843

[sagetrac-git]

Changed commit from e43e03b to 22923ab

[sagetrac-git]

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

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)
a527726	Implement diff. manifolds (tensor fields, #18843) on the new manifold categories (#18175)
22923ab	Implement diff. manifolds (tangent spaces, #19092) on the new manifold categories (#18175).

[sagetrac-git]

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

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
0ee4c41	Tensor fields on differentiable manifolds without unique representation
e8f11ff	Fix pickling test in tensor field modules.
a967787	Suppress direct call to `_element_constructor_` in tensor field parent classes
1bf77fd	Tangent spaces on differentiable manifolds without unique representation

[sagetrac-git]

Changed commit from 22923ab to 1bf77fd

[sagetrac-git]

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

d8397c1	Merge branch 'public/manifolds/top_manif_basics' of trac.sagemath.org:sage into public/manifolds/top_manif_basics
0b08b11	Some small tweaks as part of the review.
d3e5d4d	Revert to UniqueRepresentation for topological manifolds
2481359	Revert to UniqueRepresentation for ScalarFieldAlgebra; better ScalarField constructor
0cc06e8	Revert to UniqueRepresentation for topological manifold homsets
322d5bd	Revert to UniqueRepresentation for differentiable manifolds
45beae5	Revert to UniqueRepresentation for differentiable manifolds: tensor fields
98c6036	Revert to UniqueRepresentation for differentiable manifolds: tangent spaces

[sagetrac-git]

Changed commit from 1bf77fd to 98c6036

[sagetrac-git]

Changed commit from 98c6036 to 347a9cc

[sagetrac-git]

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)
0d68f86	Diff. manifolds: tensor fields, with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)
347a9cc	Diff. manifolds: tangent spaces, with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)

[sagetrac-git]

Changed commit from 347a9cc to db99641

[sagetrac-git]

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

1c1fccc	Parallel plot for Manifold: only 2D. Flat parallelization.
9b87ff1	Vectorfields : plot parallelization improvement and bug corrction.
db99641	Parallel plotting vectorfield: added example

[sagetrac-git]

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

fb7f4dd	Bibliographic references for tensor fields moved to the master file
785300b	Bibliographic references for tangent spaces moved to the master file created by #21454

[sagetrac-git]

Changed commit from 785300b to 567b892

[sagetrac-git]

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

4ce1118	Merge branch 'public/manifolds/diff_manif_tangent_spaces' of git://trac.sagemath.org/sage into public/manifolds/diff_manif_tangent_spaces
567b892	Doing some reviewer changes for #19092.

[tscrim]

Reviewer: Travis Scrimshaw

[tscrim]

comment:29

Okay, I've made it through this one. Mostly it is documentation format changes, but the exception:

For plotting, I used the @options for scale and max_range as these seemed like something a user might want to not have to pass every single time (you change change their default with v.plot.options). From the user perspective, nothing really changes unless you explicitly set said default value.

So if you agree with my changes, then you can set a positive review.

[tscrim]

comment:30

I'm handling rebasing #19124 as I'm starting my review of that now.

[sagetrac-git]

Changed commit from 567b892 to 05f8cc7

[sagetrac-git]

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

05f8cc7

A little bit of extra cleanup.

[sagetrac-git]

Changed commit from 05f8cc7 to b4a4530

[sagetrac-git]

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

23eb9d8	Correct doctest and documentation in tangent spaces
b4a4530	Use sphinx_plot for 3D plots in the documentation of charts and vector fields

[egourgoulhon]

comment:33

Thank you so much for the review!

In the first commit above (23eb9d8), I've corrected a doctest error and fixed some documentation issue revealed by the patchbot.

In the second commit (b4a4530), I've added sphinx directives to render the 3D plots in the documentation of VectorField.plot (as well as in RealChart.plot), on the same footing as what is now done in the "3D Graphics" section of the reference manual.

Thanks for the addition of @options to VectorField.plot; I've added color to these options, to be consistent with what is done in RealChart.plot. I've also added some documentation about the use of v.plot.options.

A question: in the keyword arguments of VectorField.plot, you have changed nb_values to number_values, but it is still nb_values in RealChart.plot. For consistency, both plot methods should use the same name. It seems that nb is not a standard English abbreviation for number, is it? (I find number_values a little bit long, especially that we don't have the TAB completion to type the options).

[tscrim]

comment:34

Replying to @egourgoulhon:

Thank you so much for the review!

Not a problem; sorry it took so long.

A question: in the keyword arguments of VectorField.plot, you have changed nb_values to number_values, but it is still nb_values in RealChart.plot. For consistency, both plot methods should use the same name. It seems that nb is not a standard English abbreviation for number, is it? (I find number_values a little bit long, especially that we don't have the TAB completion to type the options).

Not to my knowledge. The standard abbreviation is "no." (with a period) but "num" has become standard (at least in the programming world), but IMO the verbosity is better. However, you can do tab completion for explicit arguments.

[egourgoulhon]

comment:35

Replying to @tscrim:

Not to my knowledge. The standard abbreviation is "no." (with a period) but "num" has become standard (at least in the programming world), but IMO the verbosity is better. However, you can do tab completion for explicit arguments.

OK, so let use number_values everywhere. I am going to change it in RealChart.plot.

[tscrim]

comment:36

Thanks.

[sagetrac-git]

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

ff6eb09

Name of argument 'nb_values' in RealChart.plot changed to 'number_values'.

[sagetrac-git]

Changed commit from b4a4530 to ff6eb09

[egourgoulhon]

comment:38

Change nb_values --> number_values done.

[tscrim]

comment:39

I'm sorry I missed that on the previous ticket. LGTM. Thank you.

[egourgoulhon]

comment:40

Thanks for the review!

[vbraun]

Changed branch from public/manifolds/diff_manif_tangent_spaces to ff6eb09