SageManifolds metaticket 1

[egourgoulhon]

This is the implementation of manifolds resulting from the SageManifolds project, developed by the following authors.

This metaticket covers the implementation up to Sage 9.2. See the metaticket #30525 for Sage >= 9.3.

Algebraic part

• Tensors on free modules of finite rank: Tensors on free modules of finite rank #15916 (merged in Sage 6.6)
• Parallelization of tensor computations on free modules of finite rank: Parallelization of computations on tensors on free modules #18100 (merged in Sage 6.10)
• Improve category for finite rank free modules and provide list functionality for basis: Improve category for finite rank free modules and provide list functionality for basis #20770 (merged in Sage 7.3.beta3)
• Exterior powers of free modules of finite rank: Exterior powers of free modules of finite rank #23207 (merged in Sage 8.1.beta0)
• More functionalities for bases of free modules of finite rank: More functionalities for bases of free modules and vector frames #24792 (merged in Sage 8.2.beta8)
• Better treatment of the zero element: Tensor Fields: Better Zero Treatment #28562 (merged in Sage 9.0.beta3)
• Multiple symmetries and multiple contractions with index notation: Adding multiple symmetries and multiple contractions to tensors #28784 (merged in Sage 9.0.beta8)
• Tensor addition with index notation: _add_ method for tensors with indices #28787 (merged in Sage 9.1.beta2)
• Index notation for tensors should allow for unicode characters: Index notation for tensors should allow for unicode characters #29248 (merged in Sage 9.2.beta9)
• Basis-dependent isomorphism from FiniteRankFreeModule to an object in the category ModulesWithBasis: Basis-dependent isomorphism from FiniteRankFreeModule to an object in the category ModulesWithBasis #30094 (merged in Sage 9.2.beta7)
• ModulesWithBasis.linear_combination should be a method of Modules: ModulesWithBasis.linear_combination should be a method of Modules #30179 (merged in Sage 9.2.beta7)
• Immutable elements of FreeModuleTensor: Immutable elements of FreeModuleTensor #30181 (merged in Sage 9.2.beta8)
• FiniteRankFreeModule needs __classcall__: FiniteRankFreeModule needs __classcall__ #30169 (merged in Sage 9.2.beta8)
• Extend FreeModule factory to construction of FiniteRankFreeModule and CombinatorialFreeModule: Extend FreeModule factory to construction of FiniteRankFreeModule and CombinatorialFreeModule #30194 (merged in Sage 9.2.beta8)
• FiniteRankFreeModule: Simplify unique representation code for dependent modules: FiniteRankFreeModule: Simplify unique representation code for dependent modules #30250 (merged in Sage 9.2.beta8)
• Simplify _repr_ of ExtPowerFreeModule and ExtPowerDualFreeModule: ExtPowerFreeModule, ExtPowerDualFreeModule: Simplify _repr_ #30251 (merged in Sage 9.2.beta9)
• TensorFreeModule._an_element_: Create a default basis in the base module if necessary: TensorFreeModule._an_element_: Create a default basis in the base module if necessary #30254 (merged in Sage 9.2.beta8)
• FiniteRankFreeModule: Move all module identifications to methods exterior_power, dual_exterior_power, tensor_module: FiniteRankFreeModule: Move all module identifications to methods exterior_power, dual_exterior_power, tensor_module #30255 (merged in Sage 9.2.beta8)
• sage.tensor.modules.free_module_basis: Add testsuite: sage.tensor.modules.free_module_basis: Add testsuite #30287 (merged in Sage 9.2.beta8)

Topological and differential part

• Topological manifolds (over R, C or a topological field K):
  • basics (charts, subsets): Topological manifolds: basics #18529 (merged in Sage 7.1.beta1)
  • scalar fields (continuous functions to the base field): Topological manifolds: scalar fields #18640 (merged in Sage 7.3.beta0)
  • morphisms (continuous maps between manifolds): Topological manifolds: morphisms #18725 (merged in Sage 7.3.beta0)
  • SymPy as an alternative to SR for symbolic calculus on manifolds: SymPy as optional symbolic method for manifolds #22801 (merged in Sage 8.2.beta0)
  • periodic coordinates on manifods: Introduce periodic coordinates on manifolds #27548 (merged in Sage 8.8.beta0)
  • customizing simplifications in coordinate calculus: Customizing simplifications in symbolic calculus on manifolds #27601 (merged in Sage 8.8.beta1)
  • scalar field restrictions: Scalar Field Restrictions #28554 (merged in Sage 9.0.beta4)
  • add manifold orientability: Manifolds: add orientability #30178 (merged in Sage 9.2.beta9)
  • Immutability for scalar fields: Immutability for scalar fields #30266 (merged in Sage 9.2.beta8)
  • Coercion via restriction of chart functions: Coercion via restriction of chart functions #30267 (merged in Sage 9.2.beta8)
  • Scalar Field Arithmetics: Trivial Cases: Scalar Field Arithmetics: Trivial Cases #30291 (merged in Sage 9.2.beta8)
• Differentiable manifolds (over R, C or a non-discrete topological field K):
  • basics (charts, transition maps, scalar fields, morphisms): Differentiable manifolds: basics #18783 (merged in Sage 7.3.beta2)
  • vector fields, tensor fields and p-forms: Differentiable manifolds: vector fields and tensor fields #18843 (merged in Sage 7.5.beta1)
  • tangent spaces: Differentiable manifolds: tangent spaces #19092 (merged in Sage 7.5.beta3)
  • sets of vector fields as Lie algebroid: Modules of vector fields as Lie algebroids #20771 (merged in Sage 7.5.beta3)
  • curves: Curves in differentiable manifolds #19124 (merged in Sage 7.5.beta3)
  • affine connections: Affine connections on smooth manifolds #19147 (merged in Sage 7.5.beta4)
  • parallelization of Lie derivative computations: Parallelize Lie derivative computation #22200 (merged in Sage 7.6.beta3)
  • Multivector fields and the Schouten-Nijenhuis bracket: Multivector fields and the Schouten-Nijenhuis bracket #23429 (merged in Sage 8.1.beta8)
  • Deprecate CoordinatePatch and DifferentialForm classes: Deprecate CoordinatePatch and DifferentialForm to keep a single version of differential forms #24444 (merged in Sage 8.2.beta3)
  • Embedded submanifolds: Embedded submanifolds #25164 (merged in Sage 8.3.beta4)
  • Tensor fields along a submanifold: Tensor fields along a submanifold #25254 (merged in Sage 8.3.beta2)
  • Tensor series expansion: Tensor series expansion #25866 (merged in Sage 8.8.beta0)
  • Add manifold catalog: Add manifold catalog #25869 (merged in Sage 8.5.beta0)
  • Initializing the components of a tensor field while declaring it: Initializing the components of a tensor field while declaring it #27581 (merged in Sage 8.8.beta1)
  • Mixed differential forms: Characteristic Classes - MixedAlgebra #27584 (merged in Sage 8.8.beta3)
  • Ease the display of tensor fields in a coordinate frame: Ease the display of tensor fields in a coordinate frame #27655 (merged in Sage 8.8.beta3)
  • Mixed forms improvements: Mixed Forms Code Improvements #28578 (merged in Sage 9.0.beta5)
  • Vector bundles: Vector Bundles #28159 (merged in Sage 9.0.beta3)
  • Characteristic classes on vector bundles: Characteristic Classes on Vector Bundles #27784 (merged in Sage 9.0.beta8)
  • Tensor fields: consistent naming: Tensor Fields and Sections: Naming Consistencies #28564 (merged in Sage 9.0.beta7)
  • Immutability of tensor fields and mixed forms: Metaticket - Tensor Fields, Scalar Fields and Mixed Forms #28519 (metaticket)
  • Constructing a vector frame from a family of vector fields: Construction of a vector frame from a family of vector fields #28716 (merged in Sage 9.0.beta8)
  • Apply a function to all components of a tensor field: Apply a function to all components of a tensor field #29244 (merged in Sage 9.1.beta8)
  • Let diff() act on chart functions and differentiable forms: Let diff() act on chart functions and differentiable forms #29953 (merged in Sage 9.2.beta4)
  • Action for bundle connections: Action for Bundle Connections #30209 (merged in Sage 9.2.beta9)
  • Declare sections with name only: Declare Sections with Name only #30228 (merged in Sage 9.2.beta7)
  • List assignment for bundle connections: List Assignment for Bundle Connections #30208 (merged in Sage 9.2.beta8)
  • Immutability of tensor fields: Immutability of Tensor Fields #30274 (merged in Sage 9.2.beta8)
  • Add copy() method to affine connections: Affine Connection with Copy #30285 (merged in Sage 9.2.beta8)
  • Immutability of affine connections: Immutability of Affine Connections #30280 (merged in Sage 9.2.beta8)
  • Immutability for sections: Immutability for Sections #30288 (merged in Sage 9.2.beta8)
• Pseudo-Riemannian manifolds:
  • pseudo-Riemannian metrics: Pseudo-Riemannian metrics on smooth manifolds #19209 (merged in Sage 7.5.beta4)
  • Schouten, Cotton, and Cotton-York tensors: Schouten, Cotton, and Cotton-York tensors of a riemannian metric #19823 (merged in Sage 7.5.beta4)
  • integrated curves and geodesics: Integrated curves and geodesics on manifolds #22951 (merged in Sage 8.1.beta2)
  • pseudo-Riemannian manifolds: Pseudo-Riemannian manifods #24622 (merged in Sage 8.2.rc0)
  • Euclidean spaces and vector calculus: Euclidean spaces and vector calculus #24623 (merged in Sage 8.3.beta0)
  • Geometry of pseudo-Riemannian submanifolds: Geometry of pseudo-Riemannian submanifolds #25438 (merged in Sage 8.3.beta8)
  • Degenerate manifolds and submanifolds: Degenerate Metric and Degenerate Metric Manifold #26355 (merged in Sage 9.1.beta0)
  • Degenerate submanifolds: new features: Degenerate Metric Submanifold: New features #29080 (merged in Sage 9.1.beta4)
  • Tutorial about vector calculus: Tutorial about vector calculus #27703 (merged in Sage 8.9.beta0)
  • More control on the numerical ODE solver for integrated curves and geodesics: More control on the numerical ODE solver for integrated curves and geodesics #28707 (merged in Sage 9.0.beta6)
  • Add EuclideanSpace to category of metric spaces: Rename MetricSpaces parent method metric to metric_function, add EuclideanSpace to category of metric spaces #30062 (merged in Sage 9.2.beta7)
  • Dot and cross products along a differentiable map: Dot and cross products along a differentiable map #30318 (merged in Sage 9.2.beta8)
• Lie groups:
  • Nilpotent Lie groups: Nilpotent Lie groups #26344 (merged in Sage 8.4.rc0)

Bug fixes and performance improvements

• List functionality of free module bases: Fix bug in list functionality of free module bases #22518 (merged in Sage 7.6.rc0)
• Display of tensors on free modules of finite rank: Fix display of tensors on free modules of finite rank #22520 (merged in Sage 7.6.rc0)
• Checking validity of coordinate values on a chart: Fix bug in checking validity of coordinate values on a chart #22535 (merged in Sage 7.6.rc0)
• Symbolic derivatives in simplification of coordinate functions: Fix treatment of symbolic derivatives in simplification of coordinate functions #22503 (merged in Sage 7.6.rc0)
• Pullback on parallelizable manifolds: Fix treatment of pullback on parallelizable manifolds #22563 (merged in Sage 8.0.beta0)
• Tensor field restrictions on parallelizable manifolds: Fix treatment of tensor field restrictions on parallelizable manifolds #22637 (merged in Sage 8.0.beta1)
• Inverse metric on parallelizable manifolds: Fix computation of inverse metric on parallelizable manifolds #22667 (merged in Sage 8.0.beta1)
• Improvements in Jacobian determinants of transition maps: Jacobians of transition maps #22789 (merged in Sage 8.0.beta2)
• Arithmetics of coordinate functions and scalar fields without zero check of the result: Do not check for a zero result in arithmetics of coordinate functions and scalar fields #22859 (merged in Sage 8.0.beta5)
• Characteristic of coordinate function rings: Implement characteristic() for ring of coordinate functions #23329 (merged in Sage 8.1.beta2)
• Faster comparison of manifold points: Faster comparison of manifold points #23592 (merged in Sage 8.1.beta2)
• Fast comparison to zero (method is_trivial_zero()) for coordinate functions and scalar fields: Add is_trivial_zero() to coordinate functions and scalar fields on manifolds #23623 (merged in Sage 8.1.beta4)
• Minor errors in integrated curves: Minor errors in integrated curves #23838 (merged in Sage 8.1.beta6)
• Simplifications in calculus on manifolds with derivatives of symbolic functions: Simplifications in calculus on manifolds with derivatives of symbolic functions #24199 (fixed by Simplifications in calculus on manifolds via the expression tree #24232)
• Simplifications in calculus on manifolds via the expression tree: Simplifications in calculus on manifolds via the expression tree #24232 (merged in Sage 8.2.beta0)
• Better handling of matrix output for operator [:] acting on sets of tensor components: Matrix-related fixes in differential geometry #24884 (merged in Sage 8.2.beta8)
• Better restrictions on manifolds: Better restrictions on manifolds #25417 (merged in Sage 8.3.beta3)
• py3: implement __hash__ method in ContinuousMap: py3: implement __hash__ method in ContinuousMap #25502 (merged in Sage 8.3.beta5)
• Declared coordinate range not taken into account when a symbolic bound is involved: Declared coordinate range not taken into account when a symbolic bound is involved #25510 (merged in Sage 8.3.beta5)
• Improvements to integrated curves and geodesics on manifolds: Improvements to integrated curves and geodesics on manifolds #25936 (merged in Sage 8.5.beta1)
• Fix bug concerning restrictions of tensor fields: Fix bug concerning restrictions of tensorfields #26012 (merged in Sage 8.4.beta1)
• Parallelization of two functions (tensor calculus): Add parallelism for two tensor calculus functions #26168 (merged in Sage 8.5.beta5)
• Remove deprecated method view() in tensor: remove deprecated view in tensor #26392 (merged in Sage 8.4.rc0)
• py3: make all doctests passed with Python3 in src/sage/manifolds: py3: make all tests in manifolds passed #26475 (merged in Sage 8.5.beta0)
• py3: fix regression in Python3 tests in manifolds: py3: fix manifolds again #26569 (merged in Sage 8.5.beta2)
• py3: make all doctests passed with Python3 in src/sage/tensor: py3: make all tests in src/sage/tensor passed #26601 (merged in Sage 8.5.beta2)
• Remove deprecated stuff in manifolds: remove deprecated stuff in manifolds #26609 (merged in Sage 8.5.beta2)
• Make simplifications effective in the computation of the inverse of a transition map: Simplifications in the computation of the inverse of a transition map #26701 (merged in Sage 8.5.beta4)
• Bug in parallelization on manifolds: Bug in parallelization on manifolds #26777 (merged in Sage 8.5.beta6)
• Remove deprecated ScalarField's method function_chart: removed deprecated method in manifolds #27409 (merged in Sage 8.7.beta7)
• Bug in wedge of unnamed differential forms: Problem with wedge of unnamed diff forms #27576 (merged in Sage 8.8.beta1)
• Error in testing whether a scalar field belongs to a module of differential forms: Error in testing whether a scalar field belongs to a module of differential forms #27658 (merged in Sage 8.8.beta3)
• Tangent vectors should act as derivations on scalar fields: Tangent vectors should act as derivations on scalar fields #27856 (merged in Sage 8.8.rc0)
• Issue in calculus on manifolds with incomplete set of transition maps: Issue in calculus on manifolds with incomplete set of transition maps #28072 (merged in Sage 8.9.beta2)
• AttributeError when computing manifold chart after computing its vector field module: AttributeError when computing manifold chart after computing its vector field module #28331 (merged in Sage 8.9.beta7)
• Fix some issues with submanifolds and improve their documentation: Fix some issues with submanifolds and improve their documentation #28344 (merged in Sage 8.9.beta8)
• Use something instead of time() to ensure Manifold uniqueness in tests: Use something instead of time() to ensure Manifold uniqueness in tests #28365 (merged in Sage 8.9.beta8)
• Change keyword argument verbose to check in CoordChange.set_inverse() and set the default to True: Change keyword argument 'verbose' to 'check' in CoordChange.set_inverse() and set the default to True #28422 (merged in Sage 8.9.rc0)
• Bug in computing the second fundamental form of a Riemannian submanifold: Bug in computing the second fundamental form of a Riemannian submanifold #28462 (merged in Sage 8.9.rc0)
• Improve computation of connection coefficients: Improve computation of connection coefficients #28543 (merged in Sage 9.0.beta1)
• Output dicts and lists copied: Manifolds: Dicts and Lists Copied #28563 (merged in Sage 9.0.beta1)
• Fix wedge product on the right with a scalar field: Wedge Product with Scalar Fields #28579 (merged in Sage 9.0.beta1)
• Use standard notation for pullback and pushforward on manifolds: Use standard notation for pullback and pushforward on manifolds #28600 (merged in Sage 9.0.beta3)
• set_restriction behaviour when the subdomain is the domain: Tensor Fields: set_restriction Behaviour #28628 (merged in Sage 9.0.beta4)
• Unnecessary coordinate check while initializing an integrated curve: Unnecessary coordinate check while initializing an integrated curve #28669 (merged in Sage 9.0.beta5)
• Vector bundles: section module name fix: Vector Bundles: Section Module Name Fix #28690 (merged in Sage 9.0.beta5)
• Scalar field raise AttributeError on show_identifiers call: Scalar field raise AttributeError on show_identifiers call #28769 (merged in Sage 9.0.beta8)
• More flexible coercions in mixed form algebra: MixedFormAlgebra - More Flexible Coercions #28916 (merged in Sage 9.1.beta1)
• Mixed forms: zero treatment for scalar multiplication: Mixed Forms - Zero Treatment for Scalar Multiplication #28921 (merged in Sage 9.1.beta1)
• Vector bundles: documentation improvements: Vector Bundles: Documentation Improvements #28963 (merged in Sage 9.1.beta6)
• Automorphism fields: notation of inverse: Automorphism Fields: Notation of Inverse #28973 (merged in Sage 9.1.beta1)
• Bug in setting the calculus method on manifolds: Bug in setting the calculus method on manifolds #29021 (merged in Sage 9.1.beta1)
• Documentation improvements of characteristic classes: Documentation Improvements of Characteristic Classes #28854 (merged in Sage 9.1.beta6)
• Improve doctests of degenerate submanifolds: Improve doctests of Degenerate Submanifold #29440 (merged in Sage 9.1.rc0)
• Correct documentation of characteristic classes: CharClasses - Doc #29470 (merged in Sage 9.1.rc0)
• Wrong parent when using diff_form of degree zero: Wrong parent when using diff_form of degree zero #29570 (merged in Sage 9.1.rc3)
• Wedge product of multivector field with scalar field: Wedge product of multivector field with scalar field #29628 (merged in Sage 9.2.beta0)
• Bug in initialization of vector field with rational components: Bug in initialization of vector field with rational components #29639 (merged in Sage 9.2.beta0)
• Multiprocessing in tensor computations does not work under macOS: Multiprocessing in tensor computations does not work under macOS #29755
• More efficient contruction of free module bases: Speed up constructing high-dimensional Euclidean spaces #30061 (merged in Sage 9.2.beta7)
• Unable to coerce free module over Q into Category of vector spaces over Q: Basis-dependent isomorphism from FiniteRankFreeModule to an object in the category ModulesWithBasis #30094 (merged in Sage 9.2.beta7)
• Not equal operator for mixed forms: Not equal comparison for mixed forms #30108 (merged in Sage 9.2.beta7)
• Coercion from ChartFunctionRing to ScalarFieldAlgebra: Coercion from ChartFunctionRing to ScalarFieldAlgebra #30112 (merged in Sage 9.2.beta6)
• Failed conversion yields unconclusive error message: failed conversion yields unconclusive error message #30191 (merged in Sage 9.2.beta7)
• Error in display of a continuous map between open intervals: Error in display of a continuous map between open intervals #30289 (merged in Sage 9.2.beta8)
• Unions of subsets of a fixed open set X are not known to be subsets of X: Unions of subsets of a fixed open set X are not known to be subsets of X #30401 (merged in Sage 9.2.beta12)
• Replace AssertionError with ValueError when immutable elements are tried to be changed: Replace AssertionError with ValueError #30275 (merged in Sage 9.2.beta13)
• set_default_frame() too restrictive on parallelizable manifolds: set_default_frame() too restrictive on parallelizable manifolds #30320 (merged in Sage 9.2.beta13)
• Fix check of orientation in volume form: Fix check of orientation in volume form #30519 (merged in Sage 9.2.beta13)
• Some flake8 cleanup in manifolds: some flake8 cleanup in manifolds #30538 (merged in Sage 9.2.beta14)

Depends on #18175

CC: @sagetrac-mbejger @man74cio @tscrim @sagetrac-bpillet @LBrunswic @mjungmath @honglizhaobob

Component: geometry

Keywords: tensor, differential geometry

Issue created by migration from https://trac.sagemath.org/ticket/18528

[egourgoulhon]

Changed keywords from tensors, differential geometry to tensor, differential geometry

[egourgoulhon]

comment:12

All the tickets, except for #18786, are now ready for review.

[egourgoulhon]

comment:13

All the tickets are now based on the category ticket #18175, so that the manifold categories are

• Manifolds(K) for topological manifolds over a topological field K
• Manifolds(K).Differentiable() for differentiable manifolds
• Manifolds(K).Smooth() for smooth manifolds

[tscrim]

comment:14

Something I would like to see once the basics are done is a catalog of examples and common interesting manifolds:

• n-sphere
• n-torus
• real/complex projective n-space
• surfaces
• (Affine) Grassmannians
• Classical Lie groups (more for my info, a description of charts is on page 5 of https://www.dpmms.cam.ac.uk/~agk22/mfds.pdf, but this probably isn't a good atlas for doing computations)

I understand that some of these could be considered more wishlist than others. Some other wishlist items:

• Morse theory to compute homology of manifolds.
• Manifolds with boundary
• Cartesian products of manifolds (or more generally, fiber bundles)
• DeRham cohomology (see, e.g., lecture notes above)

[egourgoulhon]

comment:15

Replying to @tscrim:

Something I would like to see once the basics are done is a catalog of examples and common interesting manifolds:

Thanks for these suggestions. For sure, one should have a catalog of standard manifolds. For the time being, there are only examples available as worksheets at http://sagemanifolds.obspm.fr/examples.html, for instance

• the 2-sphere at http://sagemanifolds.obspm.fr/examples/html/SM_sphere_S2.html
• the real projective plane at http://sagemanifolds.obspm.fr/examples/html/SM_projective_plane_RP2.html
• the hyperbolic plane at http://nbviewer.ipython.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/v0.8/SM_hyperbolic_plane.ipynb

I understand that some of these could be considered more wishlist than others. Some other wishlist items:

• Morse theory to compute homology of manifolds.
• Manifolds with boundary
• Cartesian products of manifolds (or more generally, fiber bundles)
• DeRham cohomology (see, e.g., lecture notes above)

All the above seem indeed desirable extensions. Even if they are not implemented yet, we should have them in mind when setting the basics.

[egourgoulhon]

comment:16

PS: could you point to some existing catalog in Sage, in order to have some example?

[tscrim]

comment:17

• algebras.<tab> in sage/algebras/catalog.py
• crystals.<tab> in sage/combinat/crystals.catalog.py
• designs.<tab> in sage/combinat/designs.designs_catalog.py
• groups.<tab> in sage/groups/groups_catalog.py

[egourgoulhon]

comment:18

Thanks!

[kcrisman]

comment:195

Okay, great!