Construction of a vector frame from a family of vector fields

[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.

[egourgoulhon]

New commits:

​013fb8b

Add construction of a vector frame from a family of vector fields

[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.

[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.

[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...

[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.

[git]

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

​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()

[egourgoulhon]

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.

[egourgoulhon]

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

[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

[git]

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

​6635fad

Construction of a local frame on a vector bundle from a family of sections

[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.

[gh-DeRhamSource]

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

It seems, we are at beta6 now.

[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?

[git]

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

​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()

[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.

[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?

[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.

[gh-DeRhamSource]

Replying to 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.

Yeah, you're absolutely right. I thought it through once again and come to the same conclusion now. Sorry!

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

[egourgoulhon]

Thank you for the review!

[vbraun]

Merge conflict

[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)?

[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.

[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...

[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.

[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.

[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.

[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.

[egourgoulhon]

According to comment:29.