Characteristic Classes on Vector Bundles

[gh-DeRhamSource]

Characteristic classes on differentiable vector bundles.

Features

• characteristic classes on differentiable manifolds using the Chern-Weil homomorphism
• class BundleConnection introduced with rudimentary methods
• each characteristic class carries a dictionary with bundle connection as keys and mixed forms as values in order to simulate the cohomology equivalence

Planned sometime

• transgression forms

[git]

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

[git]

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

​963dede

is_field implemented

[embray]

As the Sage-8.8 release milestone is pending, we should delete the sage-8.8 milestone for tickets that are not actively being worked on or that still require significant work to move forward. If you feel that this ticket should be included in the next Sage release at the soonest please set its milestone to the next release milestone (sage-8.9).

[gh-DeRhamSource]

New commits:

​2cabe93

General structure of vector bundles implemented

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​b313f56

Characteristic Classes

[git]

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

​0a2190f

Ticket #27784: Wrong naming in Doctest

[git]

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

​1bae0ab

Trac #27784: Documentation grammar checkup

[git]

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

​32ecd26

Trac #27784: 0-th element of Pfaffian class is always zero

[gh-DeRhamSource]

I know, you are really busy right now, Eric. However, this ticket is the essence and foundation of my master thesis and I'd highly appreciate at least a short look whether the structure is okay.

Travis, could you take a short look, too?

Details can be changed later, but the rough structure should be fine before I start writing my thesis.

Thank you both! I really appreciate your effort so far. :)

[git]

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

​cacd413

Trac #27784: Pyflake errors

[git]

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

​2ec3a43

Trac #27784: Whitspace removed

[git]

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

​a78e1ec

Trac #27784: Documentation improved

[tscrim]

I won't be able to look at it for at least a few days. However, I will make it a high priority.

[gh-DeRhamSource]

Replying to tscrim:

I won't be able to look at it for at least a few days. However, I will make it a high priority.

Thank you so much! :)

[tscrim]

Here are my comments (mostly on the technical details):

Don't feel the strict need to adhere to PEP8 when it degrades readability. In particular, don't have a line break here:

-from sage.manifolds.differentiable.vector_bundle\
-                                               import DifferentiableVectorBundle
+from sage.manifolds.differentiable.vector_bundle import DifferentiableVectorBundle

and similarly here

            sage: from sage.manifolds.differentiable.bundle_connection import \
                                                                BundleConnection

In contrast, you can break .. MATH:: lines.

Remove the period/full-stop at the end here:

    - ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the bundle
      connection; if ``None``, it is set to ``name``.

I don't see the point of this note:

    .. NOTE::

        Notice that this is a *very* rudimentary form of bundle connections.
        A more detailed implementation is devoted to :trac:`28640`.

The only people who will care are developers, it is irrelevant to the use of this class, and it makes it come across as less polished.

The block after They certainly obey the structure equation:: is over-indented.

A general comment (not something you need to address here, but I noticed it now): we should have a little mix-in class for _latex_name objects (or perhaps more generally NamedObject) so we don't have so much repeated boilerplate code.

Instead of _hash_value, the more standard naming is _hash.

Error messages should follow the general Python convention:

-raise ValueError("A frame must be provided!")
+raise ValueError("a frame must be provided")

Is this relevant? # TODO: Compute transformations Should it instead raise a NotImplementedError?

Is there some way we can automate this?

        .. WARNING::

            If the connection has already forms in other frames, it is the
            user's responsibility to make sure that the 1-forms to be added
            are consistent with them.

-computation in SAGE.
+computation in Sage.

Although really I would suggest just deleting that sentence (you already are using Sage and such an internal working, if you insist on including it, should be in an ALGORITHM: section).

I generally dislike these things:

Class Documentation
-------------------

They don't convey any information and just are a bit of a distraction IMO.

_get_predefined_class is not a static method (it is just a separate function). Also `arg` -> ``arg``

This should just be an if-else statement:

    c_dict = {}
    c_dict['ChernChar'] = ('complex', 'additive', 'ch', r'\mathrm{ch}', x.exp())
    c_dict['Todd'] = ('complex', 'additive', 'Td', r'\mathrm{Td}',
                      x / (1 - (-x).exp()))
    c_dict['Chern'] = ('complex', 'multiplicative', 'c', 'c', 1 + x)
    c_dict['Pontryagin'] = ('real', 'multiplicative', 'p', 'p', 1 + x)
    c_dict['AHat'] = ('real', 'multiplicative', 'A^', r'\hat{A}',
                      x.sqrt() / (2 * (x.sqrt() / 2).sinh()))
    c_dict['Hirzebruch'] = ('real', 'multiplicative', 'L', 'L',
                            (x.sqrt() / x.sqrt().tanh()))
    c_dict['Euler'] = ('real', 'Pfaffian', 'e', 'e', x)
    # Get class from arg
    try:
        res = c_dict[arg]
    except KeyError:
        raise ValueError("the characteristic class '{}' is ".format(arg) +
                         "not predefined yet.")

unless you were hoping to cache c_dict, but I don't see the utility in that as this should not be a bottleneck.

Do not abbreviate class names and UniqueRepresentation is better being first in the MRO:

-class CharClass(SageObject, UniqueRepresentation):
+class CharacteristicClass(UniqueRepresentation, SageObject):

IMO, generally you should avoid abbreviated names for inputs/function names/etc. for the main implementation (although it is okay to have an abbreviated alias). (For another example, get_form(self, con, cmatrices=None): and use connection instead of con and def char_class to def characteristic_class) This better self-documents the code and can make things more discoverable with tab completion. For example, I might type .charact<tab> and then be surprised that nothing comes up.

In _get_coeff_list, there is no need for the coef variable (just return the result). Also

-(self._base_space._dim / 2).floor()
+self._base_space._dim // 2

and make from sage.symbolic.ring import SR at the top-level.

-- ``cmatrix`` curvature matrix
+- ``cmatrix`` -- curvature matrix

[git]

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

​47e2f51	copy method with naming
​f55b881	Doctest adapted
​f85de87	Merge branch 'develop' into t/28564/tensor_fields__consistent_naming
​eb92e6a	Doctest fixed
​5b2eee2	'copy' with individual naming
​38aa7f5	Merge branch 'develop' into t/28564/tensor_fields__consistent_naming
​9d39cb5	Merge branch 'develop' into t/28564/tensor_fields__consistent_naming
​67e5cdc	Trac #28564: One line doctest error fixed
​38eeb4a	Merge branch 't/28564/tensor_fields__consistent_naming' into char_classes-27784
​0857ec4	Trac #27784: Typos and minor code improvements

[gh-DeRhamSource]

Hopefully, I took everything into account you mentioned.

Thank you for the extensive feedback! :)

[git]

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

​1db2c89

Trac #27784: Line breaks

[egourgoulhon]

I gave a quik look. It looks nice! Thanks for having taken into account Travis' remarks. I fully agree with them. Here a few remarks from my side:

• There is a failed doctest:

  File "src/sage/manifolds/differentiable/char_class.py", line 685, in sage.manifolds.differentiable.char_class.CharacteristicClass.get_form
  Failed example:
      ch_form.display()
  Expected:
      ch(E, nabla^E) = ch_0(E, nabla^E) + zero + ch_1(E, nabla^E)
  Got:
      ch(E, nabla^E) = ch_0.0(E, nabla^E) + zero + ch_1.0(E, nabla^E)
  

• There are two warnings when generating the documentation:

  [manifolds] <unknown>:2566: DeprecationWarning: invalid escape sequence \c
  [manifolds] <unknown>:2565: DeprecationWarning: invalid escape sequence \c
  

• in the OUTPUT section of BundleConnection.add_connection_form(), a back quote is missing:

  - connection 1-form `\omega^i_j in the given frame, as an instance of
  + connection 1-form `\omega^i_j` in the given frame, as an instance of
  

• in the introduction to characteristic classes, N is not defined in the sentence

  for any continuous map `f: M \to N`
  

• in the doctest example of in DifferentiableVectorBundle.characteristic_class():

  - sage: g = M.metric('g')
  + sage: g = M.metric()
  

(since M has been defined a Lorentzian manifold, g has already been initialized, with the name 'g')

[gh-DeRhamSource]

Replying to egourgoulhon:

I gave a quik look. It looks nice! Thanks for having taken into account Travis' remarks. I fully agree with them. Here a few remarks from my side:

• There is a failed doctest:

  File "src/sage/manifolds/differentiable/char_class.py", line 685, in sage.manifolds.differentiable.char_class.CharacteristicClass.get_form
  Failed example:
      ch_form.display()
  Expected:
      ch(E, nabla^E) = ch_0(E, nabla^E) + zero + ch_1(E, nabla^E)
  Got:
      ch(E, nabla^E) = ch_0.0(E, nabla^E) + zero + ch_1.0(E, nabla^E)
  

That's strange...I don't get this error. Travis?

The rest is in progress...

[gh-DeRhamSource]

Ah, that might come from using Python 2.7. Hang on there.

[egourgoulhon]

Replying to gh-DeRhamSource:

That's strange...I don't get this error. Travis?

Are you using Python 3 Sage ?

[egourgoulhon]

Replying to gh-DeRhamSource:

Ah, that might come from using Python 2.7. Hang on there.

Yes most probably

[git]

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

​5dd53b0	Trac #28564: raw strings for LaTeX code
​2e15f7f	Merge branch 't/28564/tensor_fields__consistent_naming' into char_classes-27784
​ebbfb16	Trac #27784: manif zero forms set to unique zero + int division + documentation

[gh-DeRhamSource]

The issue about the DeprecationWarning referred to #28564. That has been fixed.

The other issues has been solved, too.

However, I'll build Sage from scratch, now.

[egourgoulhon]

With the latest version, there are two failed doctests in char_class.py. I got them on my machine and they are reported by the patchbot as well.

[git]

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

​310cf9a

Trac #27784: set_restriction placed wrong

[gh-DeRhamSource]

I guess that should be it. However, my build is not finished yet. I will start an extensive test as soon as possible.

Thanks so far! :)

[gh-DeRhamSource]

My tests did all pass. Patchbot says so, too.

By the way: Why is the patchbot yellow and not green though all tests passed?

[gh-DeRhamSource]

Replying to tscrim:

Is there some way we can automate this?

        .. WARNING::

            If the connection has already forms in other frames, it is the
            user's responsibility to make sure that the 1-forms to be added
            are consistent with them.

The concept and warning is just copied from affine_connection.py. At least it should be possible to return an error message when some incompatibilities occur on the intersection. But that's a topic for another ticket, I'd say.

[chapoton]

Patchbot says:

sage -t --long src/sage/parallel/map_reduce.py  # Timed out
sage -t --long src/doc/en/reference/references/index.rst  # Tab character found

[gh-DeRhamSource]

Replying to chapoton:

Patchbot says:

sage -t --long src/sage/parallel/map_reduce.py  # Timed out
sage -t --long src/doc/en/reference/references/index.rst  # Tab character found

The first one has nothing to do with the changes in this ticket.

However, the second has been fixed now.

Thanks! :)

[git]

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

​3fa8d85

Trac 27784: Tab character removed

[gh-DeRhamSource]

Also, it seems there is a doctest missing in map_reduce.py. What's going on there? :-/

[tscrim]

Patchbot has come back green.

[egourgoulhon]

Replying to gh-DeRhamSource:

Replying to tscrim:

Is there some way we can automate this?

        .. WARNING::

            If the connection has already forms in other frames, it is the
            user's responsibility to make sure that the 1-forms to be added
            are consistent with them.

The concept and warning is just copied from affine_connection.py. At least it should be possible to return an error message when some incompatibilities occur on the intersection. But that's a topic for another ticket, I'd say.

I agree: this could be postponed to another ticket; in particular, checking the compatibility can be computationaly demanding and one shall discuss the appropriate way to do this, with possibly an option check=True or check=False.

[egourgoulhon]

Replying to tscrim:

Patchbot has come back green.

From my side, I think we are almost done here. Two last remarks:

• in the introduction to characteristic classes, \mathfrak{g} is not defined
• could the file char_class.py be renamed characteristic_class.py? (this would improve the readability of the directory src/sage/manifolds/differentiable and would match with the name of the class CharacteristicClass)

[git]

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

​e4bb884

Trac #27784: File renamed + documentation

[gh-DeRhamSource]

I hope, I took the renaming everywhere into account...

[egourgoulhon]

Replying to gh-DeRhamSource:

I hope, I took the renaming everywhere into account...

Thanks for the changes. I am OK for a positive review. Travis?

[tscrim]

Yep, I am also good with it.

[git]

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

​3c93ab9

Trac 27784: Merge branch 'develop' into char_classes-27784

[gh-DeRhamSource]

Please give it a positive review again...

[egourgoulhon]

The 9.0.beta6 patchbot is all green.

[git]

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

​dba0660

Trac #27784: Rank defined in documentation

[gh-DeRhamSource]

That was just a very very minor thing I noticed during my thesis preparation. Sorry.

By the way: Am I allowed to switch the review status back to "positive" by myself in such cases?

[egourgoulhon]

Replying to gh-DeRhamSource:

That was just a very very minor thing I noticed during my thesis preparation. Sorry.

No problem. It's always good to improve the documentation!

By the way: Am I allowed to switch the review status back to "positive" by myself in such cases?

I don't think we have very precise rules about this. IMHO, it seems better to have an external eye, even for seemingly minor changes.

[git]

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

​a331374	Trac #27784: `g(0)=0` condition for real additive classes added in doc
​013fb8b	Add construction of a vector frame from a family of vector fields
​71b5f0f	Replace keyword argument 'from_family' by positional argument in vector_frame()
​d0ef4d7	Add construction of a local frame from a set of vector fields in TensorBundle.local_frame()
​6635fad	Construction of a local frame on a vector bundle from a family of sections
​5c9c516	Merge branch 'public/manifolds/vector_frame_from_family-28716' of git://trac.sagemath.org/sage into Sage 9.0.beta6
​81e2f60	Minor fix in the documentation of TopologicalVectorBundle.local_frame()
​6830d30	Trac #27784: Merge trac #28716 into trac #27784

[gh-DeRhamSource]

Please check the merge and doc change.

[egourgoulhon]

Just checked that everything is in order with the new depedency #28716 and that all doctests are passed.