Opened 6 years ago
Last modified 4 months ago
#18528 new task
SageManifolds metaticket — at Version 187
Reported by: | egourgoulhon | Owned by: | egourgoulhon |
---|---|---|---|
Priority: | major | Milestone: | |
Component: | geometry | Keywords: | manifold, tensor, differential geometry |
Cc: | mbejger, mmancini, tscrim, bpillet, gh-LBrunswic, gh-mjungmath, gh-honglizhaobob | Merged in: | |
Authors: | see ticket description | Reviewers: | |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | #18175 | Stopgaps: |
Description (last modified by )
This is the implementation of manifolds resulting from the SageManifolds project, developed by the following authors.
See also the recent metaticket #30139 for future developments.
Algebraic part
- Tensors on free modules of finite rank: #15916 (merged in Sage 6.6)
- Parallelization of tensor computations on free modules of finite rank: #18100 (merged in Sage 6.10)
- 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: #23207 (merged in Sage 8.1.beta0)
- More functionalities for bases of free modules of finite rank: #24792 (merged in Sage 8.2.beta8)
- Better treatment of the zero element: #28562 (merged in Sage 9.0.beta3)
- Multiple symmetries and multiple contractions with index notation: #28784 (merged in Sage 9.0.beta8)
- Tensor addition with index notation: #28787 (merged in Sage 9.1.beta2)
- Enhance symmetry handling for tensors: #28813
- Extension of admissible indices conventions for tensors with indices: #29099
- Index notation for tensors should allow for unicode characters: #29248
- Sparse iterator for tensor components: #29619
- Compacting the display of tensor product: #29820
- Parallelization of wedge product: #29796
- Basis-dependent isomorphism from
FiniteRankFreeModule
to an object in the categoryModulesWithBasis
: #30094 (merged in Sage 9.2.beta7) - sage.tensor.modules: Add
SymPowerFreeModule
: #30095 - sage.tensor.modules: Add backends using TensorFlow Core and PyTorch: #30096
- Add category
FreeModules
(without distinguished basis): #30164 - Trivial coercion
Modules(QQ)(FiniteRankFreeModule(QQ, 3))
fails: #30174 ModulesWithBasis.linear_combination
should be a method ofModules
: #30179 (merged in Sage 9.2.beta7)- Immutable elements of
FreeModuleTensor
: #30181 (merged in Sage 9.2.beta8) FiniteRankFreeModule
needs__classcall__
: #30169 (merged in Sage 9.2.beta8)- Extend
FreeModule
factory to construction ofFiniteRankFreeModule
andCombinatorialFreeModule
: #30194 (merged in Sage 9.2.beta8) - Submodules of
TensorFreeModule
defined by the symmetries of aComponents
object: #30229 - Add construction methods to
FiniteRankFreeModule
andCombinatorialFreeModule
: #30235 - New implementation class
FiniteRankDualFreeModule
: #30241 - Make
ExtPowerFreeModule
a quotient ofTensorFreeModule
: #30242 FreeModuleAutomorphism
should not inherit fromFreeModuleTensor
: #30245FiniteRankFreeModule
: Simplify unique representation code for dependent modules: #30250 (merged in Sage 9.2.beta8)- Simplify
_repr_
ofExtPowerFreeModule
andExtPowerDualFreeModule
: #30251 (merged in Sage 9.2.beta9) 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 methodsexterior_power
,dual_exterior_power
,tensor_module
: #30255 (merged in Sage 9.2.beta8)- Use
_matmul_
operator (@): #30244 - Phased permutation groups: #30276
sage.tensor.modules.free_module_basis
: Add testsuite: #30287 (merged in Sage 9.2.beta8)- Arithmetic on tensor module elements, manifold objects: always return a copy: #30302
Topological and differential part
- Topological manifolds (over R, C or a topological field K):
- basics (charts, subsets): #18529 (merged in Sage 7.1.beta1)
- scalar fields (continuous functions to the base field): #18640 (merged in Sage 7.3.beta0)
- morphisms (continuous maps between manifolds): #18725 (merged in Sage 7.3.beta0)
- SymPy as an alternative to SR for symbolic calculus on manifolds: #22801 (merged in Sage 8.2.beta0)
- improve simplifications in calculus on manifolds: #24151
- periodic coordinates on manifods: #27548 (merged in Sage 8.8.beta0)
- customizing simplifications in coordinate calculus: #27601 (merged in Sage 8.8.beta1)
- scalar field restrictions: #28554 (merged in Sage 9.0.beta4)
- add a bit of typing to manifold code: #29775
- add manifold orientability: #30178 (merged in Sage 9.2.beta9)
- Coercion/conversion
SR
toScalarFieldAlgebra
: #30232 - Immutability for scalar fields: #30266 (merged in Sage 9.2.beta8)
- Coercion via restriction of chart functions: #30267 (merged in Sage 9.2.beta8)
- Scalar Field Arithmetics: Trivial Cases: #30291 (merged in Sage 9.2.beta8)
- Immutability of chart functions: #30310
- Differentiable manifolds (over R, C or a non-discrete topological field K):
- basics (charts, transition maps, scalar fields, morphisms): #18783 (merged in Sage 7.3.beta2)
- vector fields, tensor fields and p-forms: #18843 (merged in Sage 7.5.beta1)
- tangent spaces: #19092 (merged in Sage 7.5.beta3)
- sets of vector fields as Lie algebroid: #20771 (merged in Sage 7.5.beta3)
- curves: #19124 (merged in Sage 7.5.beta3)
- affine connections: #19147 (merged in Sage 7.5.beta4)
- parallelization of Lie derivative computations: #22200 (merged in Sage 7.6.beta3)
- Multivector fields and the Schouten-Nijenhuis bracket: #23429 (merged in Sage 8.1.beta8)
- Deprecate CoordinatePatch and DifferentialForm classes: #24444 (merged in Sage 8.2.beta3)
- Embedded submanifolds: #25164 (merged in Sage 8.3.beta4)
- Tensor fields along a submanifold: #25254 (merged in Sage 8.3.beta2)
- Tensor series expansion: #25866 (merged in Sage 8.8.beta0)
- Add manifold catalog: #25869 (merged in Sage 8.5.beta0)
- Initializing the components of a tensor field while declaring it: #27581 (merged in Sage 8.8.beta1)
- Mixed differential forms: #27584 (merged in Sage 8.8.beta3)
- Ease the display of tensor fields in a coordinate frame: #27655 (merged in Sage 8.8.beta3)
- Mixed forms improvements: #28578 (merged in Sage 9.0.beta5)
- Vector bundles: #28159 (merged in Sage 9.0.beta3)
- Characteristic classes on vector bundles: #27784 (merged in Sage 9.0.beta8)
- Tensor fields: consistent naming: #28564 (merged in Sage 9.0.beta7)
- Immutability of tensor fields and mixed forms: #28519 (metaticket)
- Automorphism fields: remove SR enforcement when computing the inverse: #28629
- Vector-valued forms: #28640
- Constructing a vector frame from a family of vector fields: #28716 (merged in Sage 9.0.beta8)
- Inheritance structure: tensor fields as sections of vector bundles: #29234
- Apply a function to all components of a tensor field: #29244 (merged in Sage 9.1.beta8)
- New algorithm for characteristic classes: #29581
- Let
diff()
act on chart functions and differentiable forms: #29953 (merged in Sage 9.2.beta4) - Characteristic classes: sequences: #30211
- Action for bundle connections: #30209 (merged in Sage 9.2.beta9)
- Declare sections with name only: #30228 (merged in Sage 9.2.beta7)
- List assignment for bundle connections: #30208 (merged in Sage 9.2.beta8)
- Immutability for manifold objects: #30261
- Mixed forms:
set_comp
,comp
: #30272 - Immutability of tensor fields: #30274 (merged in Sage 9.2.beta8)
- Add
copy()
method to affine connections: #30285 (merged in Sage 9.2.beta8) - Immutability of affine connections: #30280 (merged in Sage 9.2.beta8)
- Immutability for sections: #30288 (merged in Sage 9.2.beta8)
- Complex and almost complex manifolds:
- almost complex structures through Hodge structures: #18786
- Pseudo-Riemannian manifolds:
- pseudo-Riemannian metrics: #19209 (merged in Sage 7.5.beta4)
- Schouten, Cotton, and Cotton-York tensors: #19823 (merged in Sage 7.5.beta4)
- integrated curves and geodesics: #22951 (merged in Sage 8.1.beta2)
- pseudo-Riemannian manifolds: #24622 (merged in Sage 8.2.rc0)
- Euclidean spaces and vector calculus: #24623 (merged in Sage 8.3.beta0)
- Geometry of pseudo-Riemannian submanifolds: #25438 (merged in Sage 8.3.beta8)
- Declaring a metric along with its components: #25457
- Energy momentum tensor, Einstein tensor and Einstein space-time: #25500
- Degenerate manifolds and submanifolds: #26355 (merged in Sage 9.1.beta0)
- Degenerate submanifolds: new features: #29080 (merged in Sage 9.1.beta4)
- Tutorial about vector calculus: #27703 (merged in Sage 8.9.beta0)
- 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: #30062 (merged in Sage 9.2.beta7) - Add sphere and hyperbolic space permanently: #30189
- Dot and cross products along a differentiable map: #30318 (merged in Sage 9.2.beta8)
- Poisson manifolds:
- Add symplectic structures: #30362
- Lie groups:
- Nilpotent Lie groups: #26344 (merged in Sage 8.4.rc0)
Bug fixes and performance improvements
- List functionality of free module bases: #22518 (merged in Sage 7.6.rc0)
- Display of tensors on free modules of finite rank: #22520 (merged in Sage 7.6.rc0)
- Checking validity of coordinate values on a chart: #22535 (merged in Sage 7.6.rc0)
- Symbolic derivatives in simplification of coordinate functions: #22503 (merged in Sage 7.6.rc0)
- Pullback on parallelizable manifolds: #22563 (merged in Sage 8.0.beta0)
- Tensor field restrictions on parallelizable manifolds: #22637 (merged in Sage 8.0.beta1)
- Inverse metric on parallelizable manifolds: #22667 (merged in Sage 8.0.beta1)
- Improvements in Jacobian determinants of transition maps: #22789 (merged in Sage 8.0.beta2)
- Arithmetics of coordinate functions and scalar fields without zero check of the result: #22859 (merged in Sage 8.0.beta5)
- Characteristic of coordinate function rings: #23329 (merged in Sage 8.1.beta2)
- 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: #23623 (merged in Sage 8.1.beta4) - Minor errors in integrated curves: #23838 (merged in Sage 8.1.beta6)
- Simplifications in calculus on manifolds with derivatives of symbolic functions: #24199 (fixed by #24232)
- Simplifications in calculus on manifolds via the expression tree: #24232 (merged in Sage 8.2.beta0)
- Make simplifications in SymPy calculus on manifolds rely on SymPy only: #24290
- Better handling of matrix output for operator
[:]
acting on sets of tensor components: #24884 (merged in Sage 8.2.beta8) - Better restrictions on manifolds: #25417 (merged in Sage 8.3.beta3)
- py3: implement
__hash__
method inContinuousMap
: #25502 (merged in Sage 8.3.beta5) - 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: #25936 (merged in Sage 8.5.beta1)
- Fix bug concerning restrictions of tensor fields: #26012 (merged in Sage 8.4.beta1)
- Parallelization of two functions (tensor calculus): #26168 (merged in Sage 8.5.beta5)
- Remove deprecated method view() in tensor: #26392 (merged in Sage 8.4.rc0)
- py3: make all doctests passed with Python3 in
src/sage/manifolds
: #26475 (merged in Sage 8.5.beta0) - py3: fix regression in Python3 tests in manifolds: #26569 (merged in Sage 8.5.beta2)
- py3: make all doctests passed with Python3 in
src/sage/tensor
: #26601 (merged in Sage 8.5.beta2) - 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: #26701 (merged in Sage 8.5.beta4)
- Bug in parallelization on manifolds: #26777 (merged in Sage 8.5.beta6)
- Remove deprecated
ScalarField
's methodfunction_chart
: #27409 (merged in Sage 8.7.beta7) - Bug in parallelized computations involving symbolic functions: #27492
- Bug in wedge of unnamed differential forms: #27576 (merged in Sage 8.8.beta1)
- 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: #27856 (merged in Sage 8.8.rc0)
- 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: #28331 (merged in Sage 8.9.beta7)- 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: #28365 (merged in Sage 8.9.beta8) - Change keyword argument
verbose
tocheck
inCoordChange.set_inverse()
and set the default toTrue
: #28422 (merged in Sage 8.9.rc0) - Bug in computing the second fundamental form of a Riemannian submanifold: #28462 (merged in Sage 8.9.rc0)
- Improve computation of connection coefficients: #28543 (merged in Sage 9.0.beta1)
- Output dicts and lists copied: #28563 (merged in Sage 9.0.beta1)
- Fix wedge product on the right with a scalar field: #28579 (merged in Sage 9.0.beta1)
- 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: #28628 (merged in Sage 9.0.beta4)- Unnecessary coordinate check while initializing an integrated curve: #28669 (merged in Sage 9.0.beta5)
- Vector bundles: section module name fix: #28690 (merged in Sage 9.0.beta5)
- Scalar field raise
AttributeError
onshow_identifiers
call: #28769 (merged in Sage 9.0.beta8) - More flexible coercions in mixed form algebra: #28916 (merged in Sage 9.1.beta1)
- Mixed forms: zero treatment for scalar multiplication: #28921 (merged in Sage 9.1.beta1)
- Vector bundles: documentation improvements: #28963 (merged in Sage 9.1.beta6)
- Automorphism fields: notation of inverse: #28973 (merged in Sage 9.1.beta1)
- Bug in setting the calculus method on manifolds: #29021 (merged in Sage 9.1.beta1)
- Documentation improvements of characteristic classes: #28854 (merged in Sage 9.1.beta6)
- Improve doctests of degenerate submanifolds: #29440 (merged in Sage 9.1.rc0)
- Correct documentation of characteristic classes: #29470 (merged in Sage 9.1.rc0)
- Wrong parent when using
diff_form
of degree zero: #29570 (merged in Sage 9.1.rc3) - Wedge product of multivector field with scalar field: #29628 (merged in Sage 9.2.beta0)
- 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: #29755
- More efficient contruction of free module bases: #30061 (merged in Sage 9.2.beta7)
- Unable to coerce free module over Q into Category of vector spaces over Q: #30094 (merged in Sage 9.2.beta7)
- Not equal operator for mixed forms: #30108 (merged in Sage 9.2.beta7)
- Coercion from
ChartFunctionRing
toScalarFieldAlgebra
: #30112 (merged in Sage 9.2.beta6) - Replace
__eq__
by_richcmp_
for manifolds: #30116 - Trivial coercion
Modules(QQ)(FiniteRankFreeModule(QQ, 3))
fails: #30174 - Failed conversion yields unconclusive error message: #30191 (merged in Sage 9.2.beta7)
declare_union
yields wrong results: #30263- Replace
AssertionError
withValueError
when immutable elements are tried to be changed: #30275 - 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: #30401
Change History (187)
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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 KManifolds(K).Differentiable()
for differentiable manifoldsManifolds(K).Smooth()
for smooth manifolds
comment:14 follow-up: ↓ 15 Changed 6 years ago by
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)
comment:15 in reply to: ↑ 14 Changed 6 years ago by
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.
comment:16 Changed 6 years ago by
PS: could you point to some existing catalog in Sage, in order to have some example?
comment:17 follow-up: ↓ 18 Changed 6 years ago by
algebras.<tab>
insage/algebras/catalog.py
crystals.<tab>
insage/combinat/crystals.catalog.py
designs.<tab>
insage/combinat/designs.designs_catalog.py
groups.<tab>
insage/groups/groups_catalog.py
comment:18 in reply to: ↑ 17 Changed 6 years ago by
Thanks!
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comment:33 follow-up: ↓ 41 Changed 5 years ago by
Would there be any interest in Kontsevich graphs, which are related to Poisson structures on manifolds from what I saw? In particular, in https://arxiv.org/abs/1702.00681, there is reference to a C++ package https://github.com/rburing/kontsevich_graph_series-cpp (with the MIT license).
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comment:41 in reply to: ↑ 33 ; follow-up: ↓ 42 Changed 5 years ago by
Replying to tscrim:
Would there be any interest in Kontsevich graphs, which are related to Poisson structures on manifolds from what I saw? In particular, in https://arxiv.org/abs/1702.00681, there is reference to a C++ package https://github.com/rburing/kontsevich_graph_series-cpp (with the MIT license).
With collaborators (independent from the above) we have developed a Sage package for calculations with Kontsevich graphs, Poisson brackets and deformation quantizations; the preliminary version will be released later this year. We certainly would like to interface our code with SageManifolds.
The main thing we would need is a SageManifolds implementation of the algebra of multivector fields (exterior algebra of the tangent bundle) and its Schouten bracket https://en.wikipedia.org/wiki/Schouten%E2%80%93Nijenhuis_bracket. Are there any plans in this direction?
comment:42 in reply to: ↑ 41 ; follow-up: ↓ 43 Changed 5 years ago by
Replying to bpym:
With collaborators (independent from the above) we have developed a Sage package for calculations with Kontsevich graphs, Poisson brackets and deformation quantizations; the preliminary version will be released later this year. We certainly would like to interface our code with SageManifolds.
Very good!
The main thing we would need is a SageManifolds implementation of the algebra of multivector fields (exterior algebra of the tangent bundle) and its Schouten bracket https://en.wikipedia.org/wiki/Schouten%E2%80%93Nijenhuis_bracket. Are there any plans in this direction?
It would certainly be easy to implement multivector fields at the level of a sequence of modules over the ring of scalar fields, in the same footing as what has been done for differential forms, cf. http://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/diff_form_module.html Each module, V^{p}(M) say, will be the set of multivector fields with a fixed degree p, i.e. the set of p-vectors. Implementing the Schouten bracket atop of this as an exterior operator V^{p}(M) x V^{q}(M) --> V^{p+q-1}(M) should not be too difficult either. But in such a setting, we do not introduce explicitely the algebra of multivector fields, which is the direct sum of all the modules V^{p}(M). Would this be an issue for you?
comment:43 in reply to: ↑ 42 ; follow-up: ↓ 44 Changed 5 years ago by
Replying to egourgoulhon:
It would certainly be easy to implement multivector fields at the level of a sequence of modules over the ring of scalar fields, in the same footing as what has been done for differential forms ...
Great! This is indeed the sort of implementation I was imagining. One would like the operations of wedge product V^{p} x V^{q} -> V^{p+q} and Schouten bracket V^{p} x V^{q} -> V^{p+q-1}. One would also like to have interior contractions with forms Omega^{p} x V^{q} -> V^{q-p} and V^{p} x Omega^{q} -> Omega^{q-p}, defined when q >= p.
But in such a setting, we do not introduce explicitely the algebra of multivector fields, which is the direct sum of all the modules V^{p}(M). Would this be an issue for you?
I don't foresee any issue.
comment:44 in reply to: ↑ 43 ; follow-up: ↓ 51 Changed 5 years ago by
Replying to bpym:
Replying to egourgoulhon:
It would certainly be easy to implement multivector fields at the level of a sequence of modules over the ring of scalar fields, in the same footing as what has been done for differential forms ...
Great! This is indeed the sort of implementation I was imagining. One would like the operations of wedge product V^{p} x V^{q} -> V^{p+q} and Schouten bracket V^{p} x V^{q} -> V^{p+q-1}. One would also like to have interior contractions with forms Omega^{p} x V^{q} -> V^{q-p} and V^{p} x Omega^{q} -> Omega^{q-p}, defined when q >= p.
This seems quite straightforward to implement. Only a matter of finding time to do it...
But in such a setting, we do not introduce explicitely the algebra of multivector fields, which is the direct sum of all the modules V^{p}(M). Would this be an issue for you?
I don't foresee any issue.
Very good!
The question is then: what is your time scale? i.e. when would you like these features to be available?
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comment:51 in reply to: ↑ 44 ; follow-up: ↓ 52 Changed 4 years ago by
Replying to egourgoulhon:
Replying to bpym:
Great! This is indeed the sort of implementation I was imagining. One would like the operations of wedge product V^{p} x V^{q} -> V^{p+q} and Schouten bracket V^{p} x V^{q} -> V^{p+q-1}. One would also like to have interior contractions with forms Omega^{p} x V^{q} -> V^{q-p} and V^{p} x Omega^{q} -> Omega^{q-p}, defined when q >= p.
This seems quite straightforward to implement. Only a matter of finding time to do it...
The pure algebraic part of this has been implemented, including the interior products, see #23207. There remains to implement the differential part, in particular the Schouten bracket. On the Wikipedia page, there is some warning: "There are two different versions, both rather confusingly called by the same name." Do we agree that the thing to implement is the bracket given by the second formula in that page?
comment:52 in reply to: ↑ 51 ; follow-up: ↓ 53 Changed 4 years ago by
Replying to egourgoulhon:
The pure algebraic part of this has been implemented, including the interior products, see #23207.
Great, thank you!
There remains to implement the differential part, in particular the Schouten bracket. On the Wikipedia page, there is some warning: "There are two different versions, both rather confusingly called by the same name." Do we agree that the thing to implement is the bracket given by the second formula in that page?
Yes, we agree.
comment:53 in reply to: ↑ 52 Changed 4 years ago by
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Replying to bpym:
Replying to egourgoulhon:
There remains to implement the differential part, in particular the Schouten bracket. On the Wikipedia page, there is some warning: "There are two different versions, both rather confusingly called by the same name." Do we agree that the thing to implement is the bracket given by the second formula in that page?
Yes, we agree.
Multivector fields and their Schouten-Nijenhuis bracket are now ready (at least for review...): #23429.
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- Milestone sage-8.8 deleted
As the Sage-8.8 release milestone is pending, we should delete the sage-8.8 milestone for tickets that are not actively being worked on or that still require significant work to move forward. If you feel that this ticket should be included in the next Sage release at the soonest please set its milestone to the next release milestone (sage-8.9).
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All the tickets, except for #18786, are now ready for review.