Such an implementation would have 2 parts: 1. *Defining the objects:* The connection 1-forms, torsion 2-forms, and curvature 2-forms are all indexed sets of differential forms. They are not tensorial, but the index labels behave in many ways like tensor components. In particular, there are "up" and "down" index versions, with particular symmetries. The case of an orthonormal basis is particularly nice, leading to "down" index antisymmetry, which it would be nice to have built in.
2. *Computing the objects:* The components of the connection 1-forms are just the Christoffel symbols, but in an arbitrary frame. So when working with explicit examples, it would be enough to be able to compute the Christoffel symbols, then use them to determine the connection forms. But this requires the ability to compute the connection in non-coordinate frames. I'll settle for an implementation of question 2. However, so far as I can tell, sage.manifolds only calculates in a coordinate basis, and the VectorFrame class doesn't do tensor derivatives. If I'm missing something here, or if there's some other known way to work in an arbitrary (especially orthonormal) basis, please let me know -- ideally with an example, such as polar coordinates in an orthonormal frame. Thank you. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
