Use standard notation for pullback and pushforward on manifolds

[gh-DeRhamSource]

At this stage, the pushforward of a vector field v along a differentiable map Phi reads Phi^*v. This contradicts the common literature where the pushforward usually writes Phi_*v.

In this ticket, a discussion is wanted and eventually a correction is performed.

[egourgoulhon]

It seems to me that in some texts, Phi^* and Phi_* are used to denote respectively the pushforward and the pullback. I'll look for some precise references.

[gh-DeRhamSource]

Yeah, it's always about conventions and historical remains...

Basically, this is not completely trustworthy, but ​here is an approach of explaination on stackexchange.

[egourgoulhon]

Replying to egourgoulhon:

It seems to me that in some texts, Phi^* and Phi_* are used to denote respectively the pushforward and the pullback. I'll look for some precise references.

Well, I could not find any such reference... Moreover, I agree that in the common literature, the pushforward is denoted by Phi_* while the pullback is denoted by Phi^*. In particular, this holds for Lee's textbooks, which we use as references in the documentation of manifolds. So I am fully OK in adopting such notation. By the way, where in the current Sage documentation have you seen the opposite convention?

[egourgoulhon]

Replying to egourgoulhon:

By the way, where in the current Sage documentation have you seen the opposite convention?

OK, I found it: it is not in the documentation but in the code of the method pushforward(), in lines 1273 and 1275 of src/sage/manifolds/differentiable/diff_map.py.

[egourgoulhon]

New commits:

​cb3c97e

Change notation of pullback and pushforward

[gh-DeRhamSource]

Looks good to me. All tests passed and the docbuild looks fine. The notebooks run fine as well. Remember, you should edit the comments in the notebook files on your homepage, too.

Should I give a positive review?

[egourgoulhon]

Replying to gh-DeRhamSource:

Looks good to me. All tests passed and the docbuild looks fine. The notebooks run fine as well. Remember, you should edit the comments in the notebook files on your homepage, too.

Should I give a positive review?

Given what you said above, I think you may ;-)