SageManifolds metaticket 1

egourgoulhon commented 9 years ago

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: #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)
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: #30094 (merged in Sage 9.2.beta7)
ModulesWithBasis.linear_combination should be a method of Modules: #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 of FiniteRankFreeModule and CombinatorialFreeModule: #30194 (merged in Sage 9.2.beta8)
FiniteRankFreeModule: Simplify unique representation code for dependent modules: #30250 (merged in Sage 9.2.beta8)
Simplify _repr_ of ExtPowerFreeModule and ExtPowerDualFreeModule: #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 methods exterior_power, dual_exterior_power, tensor_module: #30255 (merged in Sage 9.2.beta8)
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): #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)
- 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 manifold orientability: #30178 (merged in Sage 9.2.beta9)
- 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)
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)
- Constructing 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: #29244 (merged in Sage 9.1.beta8)
- Let diff() act on chart functions and differentiable forms: #29953 (merged in Sage 9.2.beta4)
- 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 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)
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)
- 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)
- Dot and cross products along a differentiable map: #30318 (merged in Sage 9.2.beta8)
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)
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 in ContinuousMap: #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 method function_chart: #27409 (merged in Sage 8.7.beta7)
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 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: #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 on show_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 to ScalarFieldAlgebra: #30112 (merged in Sage 9.2.beta6)
Failed conversion yields unconclusive error message: #30191 (merged in Sage 9.2.beta7)
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 (merged in Sage 9.2.beta12)
Replace AssertionError with ValueError when immutable elements are tried to be changed: #30275 (merged in Sage 9.2.beta13)
set_default_frame() too restrictive on parallelizable manifolds: #30320 (merged in Sage 9.2.beta13)
Fix check of orientation in volume form: #30519 (merged in Sage 9.2.beta13)
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 commented 9 years ago

Description changed:

--- 
+++ 
@@ -8,7 +8,7 @@
 ## Topological and differential part

 - **topological manifolds:** 
-  - basics (charts, subsets): *ticket not set yet*
+  - basics (charts, subsets): #18529
   - scalar fields (continuous functions to the base field): *ticket not set yet*
   - morphisms (continuous maps): *ticket not set yet*
 - **smooth manifolds:**

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -9,7 +9,7 @@

 - **topological manifolds:** 
   - basics (charts, subsets): #18529
-  - scalar fields (continuous functions to the base field): *ticket not set yet*
+  - scalar fields (continuous functions to the base field): #18640
   - morphisms (continuous maps): *ticket not set yet*
 - **smooth manifolds:**
   - basics (charts, subsets): *ticket not set yet*

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -10,12 +10,12 @@
 - **topological manifolds:** 
   - basics (charts, subsets): #18529
   - scalar fields (continuous functions to the base field): #18640
-  - morphisms (continuous maps): *ticket not set yet*
-- **smooth manifolds:**
+  - morphisms (continuous maps between manifolds): #18725
+- **differentiable manifolds:**
   - basics (charts, subsets): *ticket not set yet*
-  - scalar fields (smooth real-valued functions): *ticket not set yet*
+  - scalar fields (differentiable functions to the base field): *ticket not set yet*
   - tangent spaces: *ticket not set yet*
-  - morphisms (smooth maps): *ticket not set yet*
+  - morphisms (differentiable maps): *ticket not set yet*
   - submanifolds: *ticket not set yet*
   - tensor fields: *ticket not set yet*
   - differential forms: *ticket not set yet*

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -12,13 +12,11 @@
   - scalar fields (continuous functions to the base field): #18640
   - morphisms (continuous maps between manifolds): #18725
 - **differentiable manifolds:**
-  - basics (charts, subsets): *ticket not set yet*
-  - scalar fields (differentiable functions to the base field): *ticket not set yet*
+  - basics (charts, transition maps, scalar fields, morphisms): #18783
+  - tensor fields: *ticket not set yet*
   - tangent spaces: *ticket not set yet*
-  - morphisms (differentiable maps): *ticket not set yet*
+  - differential forms: *ticket not set yet*  
   - submanifolds: *ticket not set yet*
-  - tensor fields: *ticket not set yet*
-  - differential forms: *ticket not set yet*
   - affine connections: *ticket not set yet*
 - **pseudo-riemannian manifolds:** 
   - pseudo-riemannian metrics: *ticket not set yet*

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -1,4 +1,4 @@
-This is the implementation of manifolds (mostly smooth manifolds at the moment) resulting from the [SageManifolds project](http://sagemanifolds.obspm.fr/).
+This is the implementation of manifolds resulting from the [SageManifolds project](http://sagemanifolds.obspm.fr/).

 ## Algebraic part

@@ -13,9 +13,9 @@
   - morphisms (continuous maps between manifolds): #18725
 - **differentiable manifolds:**
   - basics (charts, transition maps, scalar fields, morphisms): #18783
-  - tensor fields: *ticket not set yet*
+  - vector fields, tensor fields and p-forms: #18843
   - tangent spaces: *ticket not set yet*
-  - differential forms: *ticket not set yet*  
+  - curves: *ticket not set yet*
   - submanifolds: *ticket not set yet*
   - affine connections: *ticket not set yet*
 - **pseudo-riemannian manifolds:**

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -14,7 +14,7 @@
 - **differentiable manifolds:**
   - basics (charts, transition maps, scalar fields, morphisms): #18783
   - vector fields, tensor fields and p-forms: #18843
-  - tangent spaces: *ticket not set yet*
+  - tangent spaces: #19092
   - curves: *ticket not set yet*
   - submanifolds: *ticket not set yet*
   - affine connections: *ticket not set yet*

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -7,18 +7,20 @@

 ## Topological and differential part

-- **topological manifolds:** 
+- **Topological manifolds** (over **R**, **C** or a topological field K):
   - basics (charts, subsets): #18529
   - scalar fields (continuous functions to the base field): #18640
   - morphisms (continuous maps between manifolds): #18725
-- **differentiable manifolds:**
+- **Differentiable manifolds** (over **R**, **C** or a non-discrete topological field K):
   - basics (charts, transition maps, scalar fields, morphisms): #18783
   - vector fields, tensor fields and p-forms: #18843
   - tangent spaces: #19092
   - curves: *ticket not set yet*
   - submanifolds: *ticket not set yet*
   - affine connections: *ticket not set yet*
-- **pseudo-riemannian manifolds:** 
+- **Complex and almost complex manifolds:** 
+  - almost complex structures through Hodge structures: #18786
+- **Pseudo-riemannian manifolds:** 
   - pseudo-riemannian metrics: *ticket not set yet*

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -15,7 +15,7 @@
   - basics (charts, transition maps, scalar fields, morphisms): #18783
   - vector fields, tensor fields and p-forms: #18843
   - tangent spaces: #19092
-  - curves: *ticket not set yet*
+  - curves: #19124
   - submanifolds: *ticket not set yet*
   - affine connections: *ticket not set yet*
 - **Complex and almost complex manifolds:**

egourgoulhon commented 9 years ago

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

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -16,8 +16,7 @@
   - vector fields, tensor fields and p-forms: #18843
   - tangent spaces: #19092
   - curves: #19124
-  - submanifolds: *ticket not set yet*
-  - affine connections: *ticket not set yet*
+  - affine connections: #19147
 - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:**

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -20,7 +20,7 @@
 - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:** 
-  - pseudo-riemannian metrics: *ticket not set yet*
+  - pseudo-riemannian metrics: #19209

egourgoulhon commented 9 years ago

comment:12

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

egourgoulhon commented 9 years ago

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 commented 9 years ago

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 commented 9 years ago

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 commented 9 years ago

comment:16

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

tscrim commented 9 years ago

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 commented 9 years ago

comment:18

Thanks!

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -21,6 +21,7 @@
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:** 
   - pseudo-riemannian metrics: #19209
+  - Schouten, Cotton, and Cotton-York tensors: #19823

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -22,6 +22,7 @@
 - **Pseudo-riemannian manifolds:** 
   - pseudo-riemannian metrics: #19209
   - Schouten, Cotton, and Cotton-York tensors: #19823
+  - Euclidean spaces: #19978

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -2,13 +2,13 @@

 ## Algebraic part

-- tensors on free modules of finite rank: #15916 (integrated in Sage 6.6)
-- parallelization of tensor computations on free modules of finite rank: #18100
+- 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)

 ## Topological and differential part

 - **Topological manifolds** (over **R**, **C** or a topological field K):
-  - basics (charts, subsets): #18529
+  - basics (charts, subsets): #18529 (merged in Sage 7.1.beta1)
   - scalar fields (continuous functions to the base field): #18640
   - morphisms (continuous maps between manifolds): #18725
 - **Differentiable manifolds** (over **R**, **C** or a non-discrete topological field K):

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -9,8 +9,8 @@

 - **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
-  - morphisms (continuous maps between manifolds): #18725
+  - 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)
 - **Differentiable manifolds** (over **R**, **C** or a non-discrete topological field K):
   - basics (charts, transition maps, scalar fields, morphisms): #18783
   - vector fields, tensor fields and p-forms: #18843

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -12,7 +12,7 @@
   - 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)
 - **Differentiable manifolds** (over **R**, **C** or a non-discrete topological field K):
-  - basics (charts, transition maps, scalar fields, morphisms): #18783
+  - basics (charts, transition maps, scalar fields, morphisms): #18783 (merged in Sage 7.3.beta2)
   - vector fields, tensor fields and p-forms: #18843
   - tangent spaces: #19092
   - curves: #19124

tscrim commented 8 years ago

Description changed:

--- 
+++ 
@@ -17,6 +17,7 @@
   - tangent spaces: #19092
   - curves: #19124
   - affine connections: #19147
+  - Lie algebras: #20771
 - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:**

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -2,8 +2,9 @@

 ## 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)
+- 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)

 ## Topological and differential part

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -16,10 +16,10 @@
   - basics (charts, transition maps, scalar fields, morphisms): #18783 (merged in Sage 7.3.beta2)
   - vector fields, tensor fields and p-forms: #18843
   - tangent spaces: #19092
+  - sets of vector fields as Lie algebroid: #20771
   - curves: #19124
   - affine connections: #19147
-  - Lie algebras: #20771
-- **Complex and almost complex manifolds:** 
+  - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:** 
   - pseudo-riemannian metrics: #19209

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -14,7 +14,7 @@
   - morphisms (continuous maps between manifolds): #18725 (merged in Sage 7.3.beta0)
 - **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
+  - vector fields, tensor fields and p-forms: #18843 (merged in Sage 7.5.beta1)
   - tangent spaces: #19092
   - sets of vector fields as Lie algebroid: #20771
   - curves: #19124

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -2,22 +2,22 @@

 ## 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)
+- 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)*

 ## 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)
+  - 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)*
 - **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
-  - sets of vector fields as Lie algebroid: #20771
-  - curves: #19124
+  - 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
   - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786

egourgoulhon commented 8 years ago

Description changed:

--- 
+++ 
@@ -18,12 +18,12 @@
   - 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
+  - affine connections: #19147 *(merged in Sage 7.5.beta4)*
   - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:** 
-  - pseudo-riemannian metrics: #19209
-  - Schouten, Cotton, and Cotton-York tensors: #19823
+  - pseudo-riemannian metrics: #19209 *(merged in Sage 7.5.beta4)*
+  - Schouten, Cotton, and Cotton-York tensors: #19823 *(merged in Sage 7.5.beta4)*
   - Euclidean spaces: #19978

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -19,7 +19,8 @@
   - 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)*
-  - **Complex and almost complex manifolds:** 
+  - parallelization of Lie derivative computations: #22200
+- **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)*

tscrim commented 7 years ago

comment:33

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).

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -12,6 +12,7 @@
   - 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)*
+  - fix bug with symbolic derivatives in simplification of coordinate functions: #22503
 - **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)*
@@ -19,7 +20,7 @@
   - 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
+  - parallelization of Lie derivative computations: #22200 *(merged in Sage 7.6.beta3)*
 - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:**

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -5,6 +5,7 @@
 - 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)*
+- Fix bug in list functionality of free module bases: #22518

 ## Topological and differential part

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -6,6 +6,7 @@
 - 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)*
 - Fix bug in list functionality of free module bases: #22518
+- Fix display of tensors on free modules of finite rank: #22520

 ## Topological and differential part

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -14,6 +14,7 @@
   - 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)*
+  - fix bug in checking validity of coordinate values on a chart: #22535
   - fix bug with symbolic derivatives in simplification of coordinate functions: #22503
 - **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)*

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -5,8 +5,6 @@
 - 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)*
-- Fix bug in list functionality of free module bases: #22518
-- Fix display of tensors on free modules of finite rank: #22520

 ## Topological and differential part

@@ -14,8 +12,6 @@
   - 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)*
-  - fix bug in checking validity of coordinate values on a chart: #22535
-  - fix bug with symbolic derivatives in simplification of coordinate functions: #22503
 - **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)*
@@ -31,5 +27,11 @@
   - Schouten, Cotton, and Cotton-York tensors: #19823 *(merged in Sage 7.5.beta4)*
   - Euclidean spaces: #19978

+## Bug fixes

+- List functionality of free module bases: #22518
+- Display of tensors on free modules of finite rank: #22520
+- Checking validity of coordinate values on a chart: #22535
+- Symbolic derivatives in simplification of coordinate functions: #22503
+- Pullback on parallelizable manifolds: #22563

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -29,9 +29,9 @@

 ## Bug fixes

-- List functionality of free module bases: #22518
-- Display of tensors on free modules of finite rank: #22520
-- Checking validity of coordinate values on a chart: #22535
-- Symbolic derivatives in simplification of coordinate functions: #22503
+- 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

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -34,4 +34,6 @@
 - 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
+- Tensor field restrictions on parallelizable manifolds: #22637
+- Inverse metric on parallelizable manifolds: #22667

9f678f49-4130-412f-9e80-9b4753f727fc commented 7 years ago

comment:41

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?

egourgoulhon commented 7 years ago

comment:42

Replying to @sagetrac-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?

9f678f49-4130-412f-9e80-9b4753f727fc commented 7 years ago

comment:43

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.

egourgoulhon commented 7 years ago

comment:44

Replying to @sagetrac-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?

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -33,7 +33,7 @@
 - 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
+- Pullback on parallelizable manifolds: #22563 *(merged in Sage 8.0.beta0)*
 - Tensor field restrictions on parallelizable manifolds: #22637
 - Inverse metric on parallelizable manifolds: #22667

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -20,6 +20,7 @@
   - 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)*
+  - improvements in Jacobian determinants of transition maps: #22789
 - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:** 
@@ -34,6 +35,6 @@
 - 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
-- Inverse metric on parallelizable manifolds: #22667
+- 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)*

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -20,7 +20,6 @@
   - 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)*
-  - improvements in Jacobian determinants of transition maps: #22789
 - **Complex and almost complex manifolds:** 
   - almost complex structures through Hodge structures: #18786
 - **Pseudo-riemannian manifolds:** 
@@ -28,7 +27,7 @@
   - Schouten, Cotton, and Cotton-York tensors: #19823 *(merged in Sage 7.5.beta4)*
   - Euclidean spaces: #19978

-## Bug fixes
+## 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)*
@@ -37,4 +36,6 @@
 - 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

85b4f6d6-bdf3-41b2-8f5a-eb6008de1f91 commented 7 years ago

Description changed:

--- 
+++ 
@@ -18,6 +18,7 @@
   - 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)*
+  - integrated curves and geodesics: #22951
   - affine connections: #19147 *(merged in Sage 7.5.beta4)*
   - parallelization of Lie derivative computations: #22200 *(merged in Sage 7.6.beta3)*
 - **Complex and almost complex manifolds:**

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -38,5 +38,5 @@
 - 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 
+- Arithmetics of coordinate functions and scalar fields without zero check of the result: #22859 *(merged in Sage 8.0.beta5)*

egourgoulhon commented 7 years ago

Description changed:

--- 
+++ 
@@ -5,6 +5,7 @@
 - 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

 ## Topological and differential part

@@ -12,6 +13,7 @@
   - 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
 - **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)*

egourgoulhon commented 7 years ago

comment:51

Replying to @egourgoulhon:

Replying to @sagetrac-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?

9f678f49-4130-412f-9e80-9b4753f727fc commented 7 years ago

comment:52

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.

sagemath / sage