Multivector fields and the Schouten-Nijenhuis bracket

[Eric Gourgoulhon]

This ticket implements multivector fields (i.e. alternating contravariant tensor fields) on differentiable manifolds, via the parent classes

• MultivectorModule: set A^p(M) of multivector fields of a degree p on a manifold M, considered as a module over the commutative algebra C^k(M) of differentiable scalar fields
• MultivectorFreeModule: set A^p(M) of multivector fields of a degree p on a parallelizable manifold M, considered as a free module of rank binomial(n,p) over C^k(M) (where n=dimM)

and the element classes

• MultivectorField: multivector field on a differentiable manifold
• MultivectorFieldParal: multivector field on a parallelizable differentiable manifold

The classes MultivectorFreeModule and MultivectorFieldParal inherit from respectively ExtPowerFreeModule and AlternatingContrTensor, which have been introduced in #23207.

The classes VectorField and VectorFieldParal inherit now from the new classes MultivectorField and MultivectorFieldParal, since a vector field is a multivector field of degree 1.

The ticket implements the exterior product A^p(M) x A^q(M) ---> A^p+q(M) (method wedge), as well as the interior products

• A^p(M) x Omega^q(M) ---> Omega^q-p(M)
• Omega^p(M) x A^q(M) ---> A^q-p(M)

for p<=q, where Omega^p(M) is the C^k(M)-module of differential forms of degree p on M (method interior_product). The ticket also implements the Schouten-Nijenhuis bracket A^p(M)xA^q(M) ---> A^p+q-1(M) (method bracket), extending the Lie bracket of vector fields.

This is a follow up of #23207 within the ​SageManifolds project (see the metaticket #18528 for an overview).

[Eric Gourgoulhon]

Last 10 new commits:

​4beecb4	Merge branch 'public/manifolds/ext_powers' of git://trac.sagemath.org/sage into Sage 8.0.rc1
​1c0f037	Change the name of modules of differential forms of degree p from Lambda^p(M) to Omega^p(M)
​a73170c	Start the implementation of multivector fields on manifolds (work in progress)
​fd4d5f4	More code regarding multivector fields on manifolds (work in progress)
​6f8bec4	Implement alternating multivector fields on manifolds (interior products in progress)
​e2dee33	Complete implementation of interior products on manifolds
​19abb49	Start implementation of Schouten-Nijenhuis bracket (work in progress)
​7398cc8	Complete implementation of Schouten-Nijenhuis bracket on parallelizable manifolds
​42c4106	Add doctests and documentation for the Schouten-Nijenhuis bracket
​3ef506f	Complete doctests and documentation for the Schouten-Nijenhuis bracket

[git]

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

​08b5dae

Slight improvement in MultivectorModule._element_constructor_

[Eric Gourgoulhon]

Following what has been done in #23623 for tensor field modules, the above commit changes the potentially expensive test

if comp == 0:
    return self.zero() 

to

if isinstance(comp, (int, Integer)) and comp == 0:
    return self.zero()

in _element_constructor_ of classes MultivectorModule and MultivectorFreeModule.

[git]

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

​bff445f

Solve merge conflict in master references file

[Eric Gourgoulhon]

The above commit (cf. comment:4) solves a merge conflict with Sage 8.1.beta5 in the master file for references (src/doc/en/reference/references/index.rst).

[Travis Scrimshaw]

A few minor comments:

Typo (appears twice):

- ``other`` -- a multivector field, `b` say

No comma here:

- ``other`` -- a multivector field, of degree `p`

Could you remove a few of these extra blanklines:

+    The Lie derivative of a 2-vector field is a 2-vector field::
+
+        sage: ab.lie_der(a)
+        2-vector field on the 3-dimensional differentiable manifold R3
+
+
+
+    """
+    def __init__(self, vector_field_module, degree, name=None,
+                 latex_name=None):

It is a little easier to read

-       resu_name = None ; resu_latex_name = None
+       resu_name = None
+       resu_latex_name = None

Do you need to use Sage's Set? It is quite a bit slower than the Python set.

For things like this:

                resu._restrictions[dom] = \
                                     dom.tensor_field_module((p,0))(rst)

it is better on 1 line and go over the 80 char/line for maintenance and readability. This is more a judgment call about the lesser of two evils.

[git]

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

​b967095

Minor corrections regarding multivector fields

[Eric Gourgoulhon]

Replying to tscrim:

Thanks for your comments; the above commit takes them into account.

Do you need to use Sage's Set?

Yes, because I need the method subsets, which does not exist for Python set.

[git]

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

​fc4ef8e

Correct typo in TensorField.__call__

[Eric Gourgoulhon]

I've just noticed a typo in the __call__ method of TensorField, as well as a redundant import in vectorfield_module.py. The above commit corrects this.

[Travis Scrimshaw]

Replying to egourgoulhon:

Replying to tscrim:

Do you need to use Sage's Set?

Yes, because I need the method subsets, which does not exist for Python set.

I am pretty sure you can use:

sage: import itertools
sage: list(itertools.combinations([1,2,3], 2))
[(1, 2), (1, 3), (2, 3)]

to iterate over all subsets of a fixed size. (There is also from sage.misc.misc import powerset for an iterator over all subsets.) I don't even think you need to convert the subsets, which are tuples, to Python sets.

[git]

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

​18df6e6

Use Python's set instead of Sage's Set in the Schouten-Nijenhuis bracket

[Eric Gourgoulhon]

Replying to tscrim:

I am pretty sure you can use:

sage: import itertools
sage: list(itertools.combinations([1,2,3], 2))
[(1, 2), (1, 3), (2, 3)]

to iterate over all subsets of a fixed size. (There is also from sage.misc.misc import powerset for an iterator over all subsets.) I don't even think you need to convert the subsets, which are tuples, to Python sets.

Thanks for the tip. The above commit implements this. I've quickly checked that this is supported by Python 3 as well. Do you confirm?

[Travis Scrimshaw]

Yep, all looks good. Thanks.

[Eric Gourgoulhon]

Thank you very much for the review!