Add a few operations from linear symplectic geometry

[tobiasdiez]

📚 Description

• Add trace of a (0,2)-tensor with respect to the symplectic form (or metric)
• Add extension of the symplectic form to a bilinear form on p-forms
• Fix hodge star for symplectic manifolds (was off by a minus sign for odd-degree forms)
• Add option for hodge star to add an additional factor of (-1)^s with s being the number of negative eigenvalues of the metric, that is,

  \alpha \wedge \star \beta = (-1)^s g(\alpha, \beta) vol_g

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[egourgoulhon]

However, now the hodge star on pseudo-Riemannian manifolds is wrong. What is your convention there?

The convention for the Hodge star on a pseudo-Riemannian manifold is detailed here.
I guess it should not be changed in order not to break backward compatibility.

[egourgoulhon]

BTW, I'm happy to see that symplectic manifolds are back!

[tobiasdiez]

However, now the hodge star on pseudo-Riemannian manifolds is wrong. What is your convention there?

The convention for the Hodge star on a pseudo-Riemannian manifold is detailed here. I guess it should not be changed in order not to break backward compatibility.

Thanks! I only added the -1 factor because I got confused by some failing tests - but they were actually failing because of other reasons. As the (-1)^s convention is quite popular, I kept it behind a toggle.

Tests pass, so this is now ready for review.

[egourgoulhon]

The difficulty in providing an example of use of using in FreeModuleTensor.trace points towards reimplementing the method trace in class TensorFieldParal, leaving FreeModuleTensor.trace without the argument using. This would also get rid of the modularity issue.

[tobiasdiez]

The difficulty in providing an example of use of using in FreeModuleTensor.trace points towards reimplementing the method trace in class TensorFieldParal, leaving FreeModuleTensor.trace without the argument using. This would also get rid of the modularity issue.

Mhh, but doesn't this complicate things by having three implementations of taking the trace?

[egourgoulhon]

The difficulty in providing an example of use of using in FreeModuleTensor.trace points towards reimplementing the method trace in class TensorFieldParal, leaving FreeModuleTensor.trace without the argument using. This would also get rid of the modularity issue.

Mhh, but doesn't this complicate things by having three implementations of taking the trace?

Code duplication can be avoided by having TensorFieldParal.trace call FreeModuleTensor.trace once the up operation is performed. The major drawback is somehow loosing the large bunch of documentation of FreeModuleTensor.trace, which is currently returned by t.trace? where t is a TensorFieldParal. The advice of Matthias would be helpful here.

[mkoeppe]

That we lose the documentation of a method from a superclass whenever we override it is a general problem, which seems to be worked around by a lot of copy-paste in the existing code base. It would be great to have a general solution for that. As doc processing for the terminal is under our control (in sage.misc.sagedoc), I think it's not too hard to come up with special markup for "include super's method documentation here". (In the HTML documentation, it could just be a hyperlink, I suppose.)

[egourgoulhon]

Thank you Matthias for your feedback. It would indeed be nice to have such a feature for the documentation.

Here, the ultimate solution would be to implement the linear algebra versions of up and down, as pointed out by Tobias in #35354 (comment). But this is beyond the scope of this ticket.

[tobiasdiez]

I've now added a test for the changes in free_module_tensor. This seems to be the easiest way out for now, since the other solution (copy & paste the whole docstring for trace from tensofield to tensorfield_paral) seems to be more of maintenance burden than providing additional value. Will implement the up/down methods for normal tensors in a follow-up PR (and maybe think about a general solution how operations from linear algebra can be lifted to tensor fields by applying them pointwise).

[github-actions]

Documentation preview for this PR is ready! 🎉
Built with commit: b439ea1

[egourgoulhon]

Thanks for your work on this!

[tobiasdiez]

Thanks for the review, Eric and Matthias!