Schouten, Cotton, and Cotton-York tensors of a riemannian metric

[pang]

Add three methods to class PseudoRiemannianMetric in order to compute Schouten, Cotton, and Cotton-York tensors. This is a follow-up of #19209 within the ​SageManifolds project (see the metaticket #18528 for an overview).

[pang]

First draft

[pang]

Last 10 new commits:

​8be9a45	Merge branch 'public/manifolds/diff_manif_metrics' of git://trac.sagemath.org/sage into #19147 (affine connections on manifolds with unique representation).
​8f0659a	Pseudo-Riemannian metrics on smooth manifolds without unique representation
​cb47399	Merge branch 'diff_manif_connections' into diff_manif_metrics to propagate minor changes in affine connections.
​58723e9	Revert to UniqueRepresentation for differentiable manifolds: metric
​2918953	Diff. manifolds: metrics, with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)
​aae566d	Improve parallelization in Components and derived classes
​98810df	Diff. manifolds: metrics, with improvements in parallelization of Components
​bd5d69f	Merge branch 'public/manifolds/sm-0.9' of git://trac.sagemath.org/sage into sagemanifolds
​14b7b6c	Merge branch 'develop' of git://github.com/sagemath/sage into sagemanifolds
​5fa71f7	Add three methods schouten, cotton, cotton_york, toghether with internal variables self._schouten, self._cotton, self._cotton_york that work analogously to the method self.ricci or self.ricci_scalar

[git]

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

​8d3ba19

Modifies doctest that threw a deprecation warning

[pang]

The tests take really long to run on my (rather old) laptop. I can't come up with examples that are meaningful and take less than 5 seconds to compute. How should I proceed?

[egourgoulhon]

Replying to pang:

The tests take really long to run on my (rather old) laptop. I can't come up with examples that are meaningful and take less than 5 seconds to compute. How should I proceed?

Since the minimal non-trivial examples have to be at least 3-dimensional, it seems quite difficult to shorten signifcantly your examples. Moreover, the time taken by them (around 10 to 20 s for each of the 3 tensors) is similar to that taken in other parts of SageManifolds, e.g. in affine connections (ticket #19147). So for the moment, I would leave it as it is...

[git]

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

​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
​1f226bd	Tensor fields with the simplified hierarchy for manifold classes
​9fa77c4	Tangent spaces with the simplified hierarchy for manifold classes
​db58a0c	Curves in diff. manifolds with the simplified hierarchy for manifold classes
​0eb9d00	Added parallelization to CompFullyAntiSym.__add__
​eb0d8b5	Affine connections with the simplified hierarchy for manifold classes
​fb043ff	Metrics on diff. manifolds with the simplified hierarchy for manifold classes
​99edbe5	Merge branch 'public/manifolds/diff_manif_metrics' into Schouten_Cotton_York.

[egourgoulhon]

In the above commit, I have simply merge the latest version of the dependency ticket #19209 (updated today) into the branch of this ticket. There is no change in your code.

[egourgoulhon]

The introduction of a class attribute for the derived objects is a good idea! However, I would suggest that its name is changed from DERIVED_OBJECTS to _derived_objects. Indeed, it seems that in Sage, capital letters are not used for attributes. Moreover, the heading underscore indicates a "private" attribute, not exposed to a tab search. Otherwise, a user searching for some derivative via g.der + TAB, might find this attribute.

Here are some remarks about each of the three tensors:

• Schouten tensor: The LaTeX formula defining the Schouten tensor in the documentation should use the same notations than in the rest of the documentation of class PseudoRiemannianMetric: Ricci should be denoted by 'Ric' (cf. method ricci) and the Ricci scalar should be denoted by 'r', not by 's' (cf. method ricci_scalar)

• Cotton tensor: The LaTeX formula defining the Cotton tensor in the documentation is not correct: it should be

   C_{ijk} = \nabla_k S_{ij} - \nabla_j S_{ik}
  

  (the Cotton tensor is antisymmetric in its last two indices). Besides, in the code, the line

  cot = nabla(s).antisymmetrize(0,2)
  

  must be replaced by

  cot = 2 * nabla(s).antisymmetrize(1,2)
  

  Indeed, the convention for the index positions of a covariant derivative is (nabla(s))_{ijk} = \nabla_k s_{ij}, i.e. the derivation index is the last one, cf. the note at ​http://sagemanifolds.obspm.fr/doc/19209/reference/manifolds/sage/manifolds/differentiable/affine_connection.html . Note also the factor 2 to compensate for the 1/2 factor introduced by the antisymmetrization. Besides, I have the impression that in the literature, the Cotton tensor is generaly defined by

  C_{ijk} = (n-2) ( \nabla_k S_{ij} - \nabla_j S_{ik} )
  

  i.e. it differs from your definition by a factor n-2, cf. for instance ​https://en.wikipedia.org/wiki/Cotton_tensor. Of course in dimension 3, both definitions coincide, but could you please check the literature for n > 3 ?

• Cotton-York tensor: Since in the rest of the documentation, $\epsilon$ stands for the Levi-Civita tensor (cf. the method volume_form), there should be no $\sqrt{\det g}$ in the LaTeX formula defining the Cotton-York tensor. Moreover, the indices seem ill-placed; the correct formula is (cf. Eq. (4) in York's paper, Phys. Rev. Lett. 26, 1656 (1971) or ​https://en.wikipedia.org/wiki/Cotton_tensor)

   CY_{ij} = \frac{1}{2} \epsilon^{kl}_{\quad i} C_{jlk} 
           = \epsilon^{kl}_{\quad i} \nabla_k S_{lj}
  

  Accordingly, in the code, the ligne

  cy = -cot.contract(0,2,eps,0,1)
  

  must be replaced by

  cy = (1/2) * eps.contract(0, 1, cot, 2, 1)
  

  Besides, the test

   if n < 3:
  

  must be replaced by

   if n != 3:
  

[pang]

It's true, your suggestions are in agreement with the literature.

[pang]

One minor thing. I realized it is not possible to change the name of the tensor after the first call. Is this relevant? Should I consider this with respect to the #19209?

[egourgoulhon]

Replying to pang:

One minor thing. I realized it is not possible to change the name of the tensor after the first call.

Can't you change the name by means of the method set_name ?

[git]

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

​b8af25e	Merge branch 'public/manifolds/diff_manif_metrics' of git://trac.sagemath.org/sage into Schouten_Cotton_York
​a6aba0e	rename PseudoRiemannianMetric.DERIVED_OBJECTS -> PseudoRiemannianMetric._derived_objects
​287236a	fix documentation of PseudoRiemannianMetric.schouten (http://trac.sagemath.org/ticket/19823#comment:12)
​d5d43ab	fix Cotton tensor, accept http://trac.sagemath.org/ticket/19823#comment:12
​accdc70	fix Cotton-York tensor, accept http://trac.sagemath.org/ticket/19823#comment:12
​8330670	Merge git://trac.sagemath.org/sage into Schouten_Cotton_York
​db7b47f	Merge commit '833067007e45133b0004271aac8cbdbdbb99a4f2' into public/manifolds/Schouten_Cotton_York
​454949a	fix text fot the cotton_york doctest
​c645bd3	minor change in the Cotton formula
​ceb53cc	a much shorter example/test for g.schouten

[git]

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
​8ba4b91	Tensor fields: solve merge conflict with Sage 7.1.beta1
​2fad9db	Correct doctest in class DiffFormParal
​b47f55f	Tangent spaces: solve merge conflict with Sage 7.1.beta1
​9a74549	Curves in diff. manifolds: solve merge conflict with Sage 7.1.beta1
​ba8ca47	Affine connections: solve merge conflict with Sage 7.1.beta1
​7c39bbb	Metrics on diff. manifolds: solve merge conflict with Sage 7.1.beta1
​9c89ed8	Schouten and Cotton tensors: solve merge conflict with Sage 7.1.beta1

[egourgoulhon]

The above commit is merely to solve a merge conflict with Sage 7.1.beta1 (there is no change in your code). This conflict was due to the merge of the latest version of #18529 into Sage 7.1.beta1.

[git]

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

​57f5fce

Minor improvements in doc of Schouten and Cotton tensors

[egourgoulhon]

Thanks for your work on this! Your code is very nice. I am setting the ticket to 'positive review', but the merge in main Sage will have to wait for the merge of the dependency ticket #19209.

[jdemeyer]

Author name please

[jdemeyer]

Replying to egourgoulhon:

the merge in main Sage will have to wait for the merge of the dependency ticket #19209.

...and #18640 and #18725 and #18783 and #18843 and #19092 and #19147. Such a deep dependency tree is scary.

[egourgoulhon]

Replying to jdemeyer:

Replying to egourgoulhon:

the merge in main Sage will have to wait for the merge of the dependency ticket #19209.

...and #18640 and #18725 and #18783 and #18843 and #19092 and #19147.

Yes, each ticket depends on the preceding one.

Such a deep dependency tree is scary.

This results from the split of SageManifolds into small tickets, to ease the review process. Note that the dependency tree is linear: each ticket depends only on the preceding one, the top being now #18640. The full tree is sketched on the metaticket #18528.

[egourgoulhon]

Can we set this back to "positive review" ?

[jdemeyer]

I don't object.

[jdemeyer]

Please fix the formatting of the references, see #20496.

[git]

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

​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
​73b4a03	Tensor fields with with coordinate functions as algebra elements
​37521ae	Tangent spaces with coordinate functions as algebra elements
​bdb91df	Curves in diff. manifolds with coordinate functions as algebra elements
​0250072	Affine connections with coordinate functions as algebra elements
​f527408	Pseudo-Riemannian metrics with coordinate functions as algebra elements
​b6b7401	Schouten and Cotton tensors with coordinate functions as algebra elements

[egourgoulhon]

Replying to jdemeyer:

Please fix the formatting of the references, see #20496.

This is done in the latest commit, along with some updates to take into account changes in #18640. I am therefore setting the ticket back to positive review, with the understanding that it can be merged in Sage only after the dependency #19209 has been merged.

[git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. Last 10 new commits:

​cfecb18	Reviewer changes and tweaks for continuous maps ticket.
​efcb618	Modify authorship for continuous maps.
​84051b0	Merge branch 'public/manifolds/diff_manif_basics' of trac.sagemath.org:sage into public/manifolds/diff_manif_basics
​97172dd	Basics of differentiable manifolds with changes in morphisms of topological manifolds
​29832ec	Use @cached_method for VectorFieldModule.identity_map() and AutomorphismFieldGroup.one()
​159afb4	Tangent spaces upon latest version of tensor fields
​1b79561	Update of OpenInterval._Hom_
​aa552cb	Affine connections on the latest #19124
​a34b35d	Pseudo-Riemannian metrics on the latest #19147
​39a1325	Schouten and Cotton tensors on the latest #19209

[egourgoulhon]

The above commit is just an update taking into account changes in the preceding tickets.

[git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. Last 10 new commits:

​3b1b794	Fix doctests in tangent spaces after the merge of #20770 in sage 7.3.beta3
​c70792e	Update tangent spaces to the latest version of #18843 (vector fields)
​0d6f47e	Merge branch 'public/manifolds/diff_manif_tangent_spaces' of trac.sagemath.org:sage into public/manifolds/lie_algebras-20771
​96f09ab	Added information about the vector field module being a Lie algebroid.
​72396dd	Merge with latest #19092 to fix doctests after the merge of #20770 in sage 7.3.beta3
​835b293	Update Lie algebroids to the latest version of #18843 (vector fields)
​edde604	Update curves to the latest version of #18843 (vector fields)
​e8e433c	Update affine connections to the latest version of #18843 (vector fields)
​61e3791	Update metrics to the latest version of vector fields (#18843)
​7dac1ad	Update of Schouten and Cotton tensors to the latest #18843 (vector fields)

[egourgoulhon]

Back to positive review since the update merely progates changes in the preceding tickets.

[git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. Last 10 new commits:

​785300b	Bibliographic references for tangent spaces moved to the master file created by #21454
​a5ab39a	Update Lie algebroids to the latest version of the dependency ticket #19092
​6af8ad2	Bibliographic references for vector field modules as Lie algebroids moved to the master file created by #21454
​cd27531	Update curves to the latest version of the dependency ticket #19092
​cdb3f28	Bibliographic references for curves on manifolds moved to the master file created by #21454
​21b468f	Update affine connections to the latest version of the dependency ticket #19092
​2113197	Bibliographic references for affine connections moved to the master file created by #21454
​41c027f	Update metrics to the latest version of the dependency ticket (#19147) + Py3 preparation
​f5dd9d5	Bibliographic references for pseudo-Riemannian metrics moved to the master file created by #21454
​a641f90	Bibliographic references for Schouten and Cotton tensors moved to the master file created by #21454

[egourgoulhon]

Setting back to positive review, since the latest commit simply solves a merge conflict with #21454.

[git]

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

​906c030	Update affine connections to the latest version of the dependency ticket #19092
​c622eb9	Update metrics on manifolds to the latest version of the dependency ticket #19147
​178c1ca	Update Schouten and Cotton tensors to the latest version of the dependency ticket #19209

[egourgoulhon]

Setting back to positive review, since the latest commit merely solves a trivial merge conflict with Sage 7.5.beta3.

[git]

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

​fdd9051

Fix the treatment of bases initialization in tangent space construction

[egourgoulhon]

Setting back to positive review since the latest commit is a trivial fix of some uncommon case in tangent space construction.