Align notation of vector fields

[mjungmath]

We explain the notation \mathfrak{X}(U,\Phi) for the space of vector fields along a map and, in particular, stretch that its only an abbreviation for \Gamma<sup>\infty(\Phi</sup>* TM).

Also include a few other fairly trivial improvements to the documentation.

CC: @tobiasdiez @tscrim @egourgoulhon

Component: manifolds

Author: Tobias Diez

Branch/Commit: public/manifolds/vectorfields_docs @ 6773244

Issue created by migration from https://trac.sagemath.org/ticket/33542

[mkoeppe]

Branch: u/mkoeppe/align_notation_of_vector_fields

[mkoeppe]

Commit: bea838f

[mkoeppe]

Author: Tobias Diez

[mkoeppe]

New commits:

bea838f

Documentation changes split out from #29775 'Add a bit of typing to manifold code'

[tobiasdiez]

comment:3

C^k(U;E)

From what I've seen, this is the usual notation for arbitrary k-smooth maps from U to E; the set of global sections of E is usually denoted by \Gamma(E) or \Gamma^k(E), and \Gamma^k(U; E) (or \Gamma^k_U(E)) for sections over a subset U of the base.

[tobiasdiez]

comment:4

\mathfrak{X}(U,\Phi)

\mathfrak{X}(M) is a pretty common notation for the set of vector fields on M. I don't think there is a standard notation for vector fields along a map, but \mathfrak{X}(U,\Phi) seems like a good convention (which however needs explanation).

[egourgoulhon]

comment:5

Replying to @tobiasdiez:

\mathfrak{X}(U,\Phi)

\mathfrak{X}(M) is a pretty common notation for the set of vector fields on M. I don't think there is a standard notation for vector fields along a map, but \mathfrak{X}(U,\Phi) seems like a good convention (which however needs explanation).

I agree: \mathfrak{X}(M) is a standard notation and we should keep it. It is then quite natural to use \mathfrak{X}(U,\Phi) for vector fields along a map.

[egourgoulhon]

comment:6

May I ask about the purpose of these doctests:

+        TESTS:
+
+        Identity map of a differentiable manifold::
+
+            sage: M = Manifold(2, 'M', structure='differentiable')
+            sage: id = M.identity_map(); id
+            Identity map Id_M of the 2-dimensional differentiable manifold M
+            sage: isinstance(id, sage.manifolds.differentiable.diff_map.DiffMap)
+            True
+
+        Identity map of a topological manifold is not smooth::
+        
+            sage: M = Manifold(2, 'M', structure='topological')
+            sage: isinstance(M.identity_map(), sage.manifolds.differentiable.diff_map.DiffMap)
+            False

They are pretty trivial and I am not sure they are useful. Moreover, I don't see the connection with the topic of this ticket.

[mjungmath]

comment:7

I agree, \mathfrak{X}(M) is common. That is not what I meant. I rephrased the description (frantically opened this ticket this morning between two meetings).

In principle, I just wanted to start a discussion as these changes were proposed by Tobias in another ticket. @Tobias, maybe you can point out the purpose of these changes that you had in mind, and modify the description accordingly?

[tobiasdiez]

comment:8

Replying to @egourgoulhon:

May I ask about the purpose of these doctests:
They are pretty trivial and I am not sure they are useful. Moreover, I don't see the connection with the topic of this ticket.

They were a result of a discussion/confusion at the beginning of #29775.

[mjungmath]

comment:10

I don't like this notation. This looks like a statement. I'd prefer something like this:

-    the *vector field module* `\mathfrak{X}(U,\Phi)` is the set of
+    the *vector field module* `\mathfrak{X}(U,\Phi)` (sometimes also denoted as 
+    `\Gamma(\Phi^* TM)` or `C^k(U; \Phi^* TM) )` is the set of

Also, it could be a nice idea to add a reference to the section modules of vector bundles.

It might be worth to check other files and make the changes accordingly.

[mjungmath]

comment:12

Moreover, I agree with Eric. I still don't see the purpose of this (trivial) test.

[tobiasdiez]

comment:13

Replying to @mjungmath:

I don't like this notation. This looks like a statement.

It is also a statement. \mathfrak{X}(M) is an abbreviation for \Gamma^\infty(TM), i.e. \mathfrak{X}(M) \equiv \Gamma^\infty(TM). I've clarified this point now.

[tobiasdiez]

comment:14

Replying to @mjungmath:

Moreover, I agree with Eric. I still don't see the purpose of this (trivial) test.

Ok, removed.

[tobiasdiez]

comment:15

(Don't have the rights to update Mathias branch on trac, but they are on github sagemath/sagetrac-mirror@develop...u/mkoeppe/align_notation_of_vector_fields

[mkoeppe]

comment:16

Just put the name of your branch in the "Branch:" field

[tobiasdiez]

Changed branch from u/mkoeppe/align_notation_of_vector_fields to public/manifolds/vectorfields_docs

[sagetrac-git]

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

955be53

Remove tests

[sagetrac-git]

Changed commit from bea838f to 955be53

[sagetrac-git]

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

6773244

Clarify docs for vector field module

[sagetrac-git]

Changed commit from 955be53 to 6773244