Affine connections on smooth manifolds

[egourgoulhon]

This ticket implements affine connections on infinitely differentiable manifolds (i.e. smooth manifolds) . This is a follow-up of #19092 within the ​SageManifolds project (see the metaticket #18528 for an overview). As in #19092, the non-discrete topological field K over which the smooth manifold is defined is generic, although in most applications, K=R or K=C.

Affine connections are implemented via the Python class AffineConnection, the user interface being the method DifferentiableManifold.affine_connection(). At the user choice, CPU-demanding computations (like that of the curvature tensor) can be parallelized, thanks to #18100.

Various methods of the class AffineConnection allow the computation of

• the connection coefficients with respect to a given vector frame (from those w.r.t. another frame)
• the connection 1-forms with respect to a given vector frame
• the torsion tensor
• the torsion 2-forms with respect to a given vector frame
• the (Riemann) curvature tensor
• the curvature 2-forms with respect to a given vector frame
• the Ricci tensor
• the action of the affine connection on any tensor field

Documentation: The reference manual is produced by sage -docbuild reference/manifolds html It can also be accessed online at ​http://sagemanifolds.obspm.fr/doc/19147/reference/manifolds/ More documentation (e.g. example worksheets) can be found ​here.

[git]

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

​477d057

Add doctests to AffineConnection; full coverage

[git]

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

​a148062

Major improvements in the documentation of diff. manifolds (affine connections part)

[git]

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

​a80dccc

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

[git]

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

​325e832	Corrected error in parallelisation of __mul__. Added parallism for the symmetric __add__
​256d268	changed name TensorParallelism to TensorParallelCompute
​bea739d	Added Eric corrections to parallelism
​a6d4bd4	Merge branch 'public/18100-parallelization_tensors' into Sage 6.7
​81f662c	Improve documentation of class TensorParallelCompute.
​ea38c84	In doc, changed processor with process
​819661d	Changed variables name : set_nproc to set_nproc_tensor, get_nproc to get_nproc_tensor
​53c248c	Parallelization: corrected doc
​0bb79a3	Merge #18100 into #19124
​f86e5db	Merge #19147 into (#19124 + #18100)

[git]

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

​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).
​666bbab	Implement diff. manifolds (curves, #19124) on the new manifold categories (#18175)
​fea227e	Implement diff. manifolds (affine connections, #19147) on the new manifold categories (#18175).

[git]

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

​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
​dca7f9d	Curves on differentiable manifolds without unique representation
​bcd87e3	Merge branch 'public/manifolds/diff_manif_curves' of git://trac.sagemath.org/sage into diff_manif_connections
​0032d27	Affine connections on differentiable manifolds without unique representation

[egourgoulhon]

The parallelization is not effective in the latest commit (cf. the discussion starting from comment 74 in #18529). Hence the FIXME comment in the doctest at line 1730 of affine_connection.py.

[git]

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

​1344776

Add method AffineConnection.__ne__

[git]

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

​26e8dae

First argument of scalar field constructor is the scalar field algebra (i.e. the parent)

[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
​b836212	Revert to UniqueRepresentation for differentiable manifolds: curves
​0ea8b0c	Revert to UniqueRepresentation for differentiable manifolds: affine connections

[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)
​6b51926	Diff. manifolds: curves, with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)
​d528d98	Diff. manifolds: affine connections, with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)

[git]

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

​aae566d

Improve parallelization in Components and derived classes

[git]

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

​0eb9d00

Added parallelization to CompFullyAntiSym.__add__

[git]

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

​b07ffd9	Tangent spaces on refactored differentiable manifolds
​3cd03a4	Add methods lift() and retract() to ManifoldSubset; add method __eq__() in CoordChange
​984c3c2	Revert to simple hierarchy for manifold classes
​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
​eb0d8b5	Affine connections with the simplified hierarchy for manifold classes

[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

[git]

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

​7b3dab3	Merge branch 'public/manifolds/top_manif_scalar_fields' of git://trac.sagemath.org/sage into Sage 7.2.rc0
​9ec7d3e	Python 3 format for print in manifolds
​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

[git]

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

​4fad094	Merge branch 'public/manifolds/top_manif_morphisms' of trac.sagemath.org:sage into public/manifolds/top_manif_morphisms
​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

[git]

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

​d7ad8e1	Merge branch 'public/manifolds/diff_manif_tensor_fields' of git://trac.sagemath.org/sage into sage 7.3.beta5
​dcb08fc	Decrease doctest times in tensor fields; remove xder from the global namespace
​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)

[git]

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

​4955c3c	Final review of everything.
​7cf4ff4	Merge branch 'public/manifolds/diff_manif_tensor_fields' of trac.sagemath.org:sage into public/manifolds/diff_manif_tensor_fields
​58ae506	Merge branch 'public/manifolds/diff_manif_tensor_fields' of git://trac.sagemath.org/sage into Sage 7.4.beta6
​8219403	Fix documentation error; change in TensorField comparison to zero
​a841f79	Python 3 compatible syntax in tensor fields
​5a5f400	Slight change in TensorField.__eq__ (comparison to zero)
​98efa66	Update tangent spaces to the final version of the dependency ticket #18843
​a5ab39a	Update Lie algebroids to the latest version of the dependency ticket #19092
​cd27531	Update curves to the latest version of the dependency ticket #19092
​21b468f	Update affine connections to the latest version of the dependency ticket #19092

[git]

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

​8860387	Add missing reference Lee13
​fb7f4dd	Bibliographic references for tensor fields moved to the master file
​8205dca	Update tangent spaces to the last change (adding a missing reference) in the dependency ticket #18843
​785300b	Bibliographic references for tangent spaces moved to the master file created by #21454
​6af8ad2	Bibliographic references for vector field modules as Lie algebroids moved to the master file created by #21454
​cdb3f28	Bibliographic references for curves on manifolds moved to the master file created by #21454
​2113197	Bibliographic references for affine connections moved to the master file created by #21454

[git]

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

​906c030

Update affine connections to the latest version of the dependency ticket #19092

[egourgoulhon]

Replying to vbraun:

Thank you!