#28716 closed enhancement (fixed)

Construction of a vector frame from a family of vector fields

Reported by: egourgoulhon Owned by:
Priority: major Milestone: sage-9.0
Component: geometry Keywords: manifolds, vector_frame
Cc: tscrim Merged in:
Authors: Eric Gourgoulhon Reviewers: Michael Jung
Report Upstream: N/A Work issues:
Branch: 81e2f60 (Commits) Commit: 81e2f60b2247b473b0d2aa8c059a2211faa4ff63
Dependencies: Stopgaps:

Description (last modified by egourgoulhon)

This ticket modifies DifferentiableManifold.vector_frame() to allow for constructing a vector frame from a spanning family of linearly independent vector fields:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: e0 = M.vector_field(1+x^2, 1+y^2)
sage: e1 = M.vector_field(2, -x*y)
sage: e = M.vector_frame('e', (e0, e1)); e
Vector frame (M, (e_0,e_1))
sage: e[0].display()
e_0 = (x^2 + 1) d/dx + (y^2 + 1) d/dy
sage: e[1].display()
e_1 = 2 d/dx - x*y d/dy
sage: (e[0], e[1]) == (e0, e1)
True

Previously, the only way to introduce the vector frame e was to first introduce the automorphism relating the frame (d/dx, d/dy) to (e0, e1) and to pass this automorphism to VectorFrame.new_frame():

sage: aut = M.automorphism_field()
sage: aut[:] = [[e0[0], e1[0]], [e0[1], e1[1]]]
sage: e = X.frame().new_frame(aut, 'e')

Implementation details: such functionality already existed for bases of finite rank free modules; the relevant code is extracted from the method FiniteRankFreeModule.basis() and put into the new method FreeModuleBasis._init_from_family(), in order to be used in DifferentiableManifold.vector_frame() as well.

Change History (31)

comment:1 Changed 13 months ago by egourgoulhon

  • Branch set to public/manifolds/vector_frame_from_family-28716
  • Commit set to 013fb8b665adaf1339372c457fca851246d30a3c

New commits:

013fb8bAdd construction of a vector frame from a family of vector fields

comment:2 Changed 13 months ago by egourgoulhon

  • Cc tscrim added
  • Status changed from new to needs_review

comment:3 Changed 13 months ago by egourgoulhon

  • Description modified (diff)

comment:4 follow-up: Changed 13 months ago by tscrim

Do we need the optional parameter? Basically, can we just use the fact that a tuple/list is being given and then assume it is suppose to be a family of vector fields? If it has to be a keyword, I would change from_family to the more descriptive from_vector_fields.

comment:5 follow-up: Changed 13 months ago by gh-DeRhamSource

This is a great idea and would be very useful for vector bundles, too. Sometimes I got really annoyed by this detour. Would you mind to adapt your code, if working, for vector bundles as-well?

By the way: We should combine vector bundles and the previous implementations really really soon (in this case inherit vector frames from local frames) otherwise things could get extremly messy.

Unfortunately, I am quite busy working at my master thesis right now. I can almost feel the deadline touching my skin. I promise to work on that as soon as I've gained some time back.

Even though I don't have the time now, I've opened the corresponding ticket #28718, just to keep this task in mind.

Last edited 13 months ago by gh-DeRhamSource (previous) (diff)

comment:6 in reply to: ↑ 4 Changed 13 months ago by egourgoulhon

Replying to tscrim:

Thanks for your prompt feedback.

Do we need the optional parameter? Basically, can we just use the fact that a tuple/list is being given and then assume it is suppose to be a family of vector fields?

Good idea, this is much more user-friendly! I am on it...

comment:7 in reply to: ↑ 5 Changed 13 months ago by egourgoulhon

Replying to gh-DeRhamSource:

This is a great idea and would be very useful for vector bundles, too. Sometimes I got really annoyed by this detour.

Yes, this should have been done sooner...

Would you mind to adapt your code, if working, for vector bundles as-well?

OK, I'll try to do this (see below).

By the way: We should combine vector bundles and the previous implementations really really soon (in this case inherit vector frames from local frames) otherwise things could get extremly messy.

Yes, I agree. Note however that this ticket does not touch the class VectorFrame, only the user interface DifferentiableManifold.vector_frame(). I'll perform a similar change to the interfaces TopologicalVectorBundle.local_frame() and TensorBundle.local_frame().

Unfortunately, I am quite busy working at my master thesis right now. I can almost feel the deadline touching my skin.

Good luck with your master thesis!

I promise to work on that as soon as I've gained some time back. Even though I don't have the time now, I've opened the corresponding ticket #28718, just to keep this task in mind.

Thanks.

comment:8 Changed 13 months ago by git

  • Commit changed from 013fb8b665adaf1339372c457fca851246d30a3c to d0ef4d77e44422e25fbdfbc90e9c852387a15033

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

71b5f0fReplace keyword argument 'from_family' by positional argument in vector_frame()
d0ef4d7Add construction of a local frame from a set of vector fields in TensorBundle.local_frame()

comment:9 Changed 13 months ago by egourgoulhon

  • Description modified (diff)

In the latest version (cf. comment:8 commits)

  • vector_frame() accepts a tuple/list of vector fields as a positional argument, the keyword argument from_family being suppressed, following the suggestion made in comment:4.
  • The ZeroDivisionError that occurs if the vector fields are not linearly independent (the exception is raised when computing the inverse of the automorphism relating the new frame to a previous one) is cached with a proper error message.
  • The documentation of the module sage.manifolds.differentiable.vectorframe has been updated to take into account the new functionality.
  • TensorBundle.local_frame() has been updated to offer the same functionality, following comment:5.
  • A TODO section has been added to TopologicalVectorBundle.local_frame() for implementing a similar functionality with local sections in the future.

I propose to stay here for this ticket, i.e. to let the modification of TopologicalVectorBundle.local_frame() to a future ticket (#28718 ?). This is mostly to avoid code duplication with DifferentiableManifold.vector_frame(), waiting for a clearer view of #28718. Besides, I will be extremely busy in the coming weeks and I would like very much the vector_frame() functionality introduced in the current ticket to make its way in Sage 9.0.

comment:10 Changed 13 months ago by egourgoulhon

Do you agree with the above changes (comment:9)?

comment:11 follow-up: Changed 13 months ago by gh-DeRhamSource

Partially. Of course, we can postpone this to another ticket, preferrably to #28718. However, I think this fits in here perfectly well and enables the feature for vector bundles in Sage 9.0 without too much effort [1] as you already did it for the tensor bundle. But I have no strong opinion on that so just do as you prefer.

One personal reformulation:

-        any connection with previously defined vector frames or vector fields
-        (the connection can be performed later via the method
+        connecting it to previously defined vector frames or vector fields
+        (this can still be performed later via the method

[1] at least so far as I can see

comment:12 Changed 12 months ago by git

  • Commit changed from d0ef4d77e44422e25fbdfbc90e9c852387a15033 to 6635fad2f2c51e5b8e99ea4da98e6b9cd26b020c

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

6635fadConstruction of a local frame on a vector bundle from a family of sections

comment:13 in reply to: ↑ 11 Changed 12 months ago by egourgoulhon

Replying to gh-DeRhamSource:

Partially. Of course, we can postpone this to another ticket, preferrably to #28718. However, I think this fits in here perfectly well and enables the feature for vector bundles in Sage 9.0 without too much effort [1] as you already did it for the tensor bundle.

OK I've done it in the above commit. I've also improved the catching of the error in case of linearly dependent elements.

One personal reformulation:

-        any connection with previously defined vector frames or vector fields
-        (the connection can be performed later via the method
+        connecting it to previously defined vector frames or vector fields
+        (this can still be performed later via the method

Done as well.

comment:14 Changed 12 months ago by gh-DeRhamSource

Thank you so much! I'll give it a further look this or tomorrow afternoon.

It seems, we are at beta6 now.

comment:15 follow-up: Changed 12 months ago by gh-DeRhamSource

I gave it some short tests. This is a huge improvement for using frames! Thanks! :)

Just a minor thing:

-          ``self`` (`n` being the dimension of ``self``) defining the local
+          ``self`` (`n` being the rank of ``self``) defining the local

Apart from that it looks fine to me. As soon as you merged the recent develop branch into this one and patchbot says "yes", I could give it a positive review. Travis?

Last edited 12 months ago by gh-DeRhamSource (previous) (diff)

comment:16 Changed 12 months ago by git

  • Commit changed from 6635fad2f2c51e5b8e99ea4da98e6b9cd26b020c to 81e2f60b2247b473b0d2aa8c059a2211faa4ff63

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

5c9c516Merge branch 'public/manifolds/vector_frame_from_family-28716' of git://trac.sagemath.org/sage into Sage 9.0.beta6
81e2f60Minor fix in the documentation of TopologicalVectorBundle.local_frame()

comment:17 in reply to: ↑ 15 Changed 12 months ago by egourgoulhon

Replying to gh-DeRhamSource:

Just a minor thing:

-          ``self`` (`n` being the dimension of ``self``) defining the local
+          ``self`` (`n` being the rank of ``self``) defining the local

Thanks for pointing this; it is corrected in the above commit.

comment:18 follow-up: Changed 12 months ago by gh-DeRhamSource

There is one thing I'm not sure about. Namely the line:

            mat = [[c[[i]] for c in comps] for i in fmodule.irange()]
            aut.add_comp(basis)[:] = mat
   this --> aut.add_comp(self)[:] = mat
            fmodule.set_change_of_basis(basis, self, aut)

in free_module_basis.py.

Shouldn't it be the identity matrix with respect to that basis? Or did I get something wrong?

comment:19 in reply to: ↑ 18 ; follow-up: Changed 12 months ago by egourgoulhon

Replying to gh-DeRhamSource:

There is one thing I'm not sure about. Namely the line:

            mat = [[c[[i]] for c in comps] for i in fmodule.irange()]
            aut.add_comp(basis)[:] = mat
   this --> aut.add_comp(self)[:] = mat
            fmodule.set_change_of_basis(basis, self, aut)

in free_module_basis.py.

Shouldn't it be the identity matrix with respect to that basis? Or did I get something wrong?

The formula is correct: it should not be the identity matrix but the matrix of the change-of-basis automorphism, which has the same expression in both bases.

comment:20 in reply to: ↑ 19 Changed 12 months ago by gh-DeRhamSource

  • Reviewers set to Michael Jung
  • Status changed from needs_review to positive_review

Replying to egourgoulhon:

The formula is correct: it should not be the identity matrix but the matrix of the change-of-basis automorphism, which has the same expression in both bases.

Yeah, you're absolutely right. I thought it through once again and come to the same conclusion now. Furthermore, some tests on this did work properly. Sorry!

So from my perspective, everything is fine. I'll give it a positive review.

Last edited 12 months ago by gh-DeRhamSource (previous) (diff)

comment:21 Changed 12 months ago by egourgoulhon

Thank you for the review!

comment:22 follow-up: Changed 12 months ago by vbraun

  • Status changed from positive_review to needs_work

Merge conflict

comment:23 in reply to: ↑ 22 ; follow-up: Changed 12 months ago by egourgoulhon

Replying to vbraun:

Merge conflict

There is no conflict with the just released 9.0.beta7. Is it a conflict with #27784 (which is not merged yet)?

comment:24 in reply to: ↑ 23 ; follow-up: Changed 12 months ago by gh-DeRhamSource

Replying to egourgoulhon:

Replying to vbraun:

Merge conflict

There is no conflict with the just released 9.0.beta7. Is it a conflict with #27784 (which is not merged yet)?

Not sure. But to run some tests for my thesis, I merged these two tickets and a conflict occured. Just a very minor thing about lines if I remember correctly.

However, either ticket needs to be merged to be certain.

comment:25 in reply to: ↑ 24 Changed 12 months ago by egourgoulhon

Replying to gh-DeRhamSource:

Replying to egourgoulhon:

Replying to vbraun:

Merge conflict

There is no conflict with the just released 9.0.beta7. Is it a conflict with #27784 (which is not merged yet)?

Not sure. But to run some tests for my thesis, I merged these two tickets and a conflict occured. Just a very minor thing about lines if I remember correctly.

Yes most of the time these conflicts due to various developments performed in parallel are very minor and easy to solve.

However, either ticket needs to be merged to be certain.

Yes. I am afraid we have to wait for the next beta to solve this...

comment:26 follow-up: Changed 12 months ago by tscrim

Since you basically know what ticket, I would just merge that in and set this back to positive review with that as a dependency.

comment:27 in reply to: ↑ 26 ; follow-up: Changed 12 months ago by gh-DeRhamSource

Replying to tscrim:

Since you basically know what ticket, I would just merge that in and set this back to positive review with that as a dependency.

Good idea. I'll do that for #27784. I need to add a minor thing into the documentation of characteristic classes anyway.

comment:28 in reply to: ↑ 27 ; follow-up: Changed 12 months ago by gh-DeRhamSource

Replying to gh-DeRhamSource:

Replying to tscrim:

Since you basically know what ticket, I would just merge that in and set this back to positive review with that as a dependency.

Good idea. I'll do that for #27784. I need to add a minor thing into the documentation of characteristic classes anyway.

Done.

comment:29 in reply to: ↑ 28 Changed 12 months ago by egourgoulhon

Replying to gh-DeRhamSource:

Replying to gh-DeRhamSource:

Replying to tscrim:

Since you basically know what ticket, I would just merge that in and set this back to positive review with that as a dependency.

Good idea. I'll do that for #27784. I need to add a minor thing into the documentation of characteristic classes anyway.

Done.

Thanks!

I am then setting this ticket back to positive review and will have a look at #27784.

comment:30 Changed 12 months ago by egourgoulhon

  • Status changed from needs_work to positive_review

According to comment:29.

comment:31 Changed 12 months ago by vbraun

  • Branch changed from public/manifolds/vector_frame_from_family-28716 to 81e2f60b2247b473b0d2aa8c059a2211faa4ff63
  • Resolution set to fixed
  • Status changed from positive_review to closed
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