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