Adding notebook: Cartan structural equations and Bianchi identities

[Doxdrum]

A short notebook on the manipulation of the structural equations, using differential forms. Inspired by an example from exterior.cnb

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Note: If a covariant exterior derivative could be defined, say $D$, the Bianchi identities can be written as $D T^a = R^a{}_b e^b$ and $D R^a{}_b = 0$. So far, I couldn't write such derivative.

[kpeeters]

You can avoid duplication of those structural equations (when you simplify the Bianchis) by e.g. using

from cdb.core.manip import *
de:= @(struc1):
isolate(de, $d{e^{a}}$);

after which those substitutes become

substitute(Bianchi1, de);

and similar for the other one.

For the D operator it all depends what you want to do next, but I guess what you are after is some way to define an operator which respects form degrees but can be expanded in terms of the ordinary d and spin connection terms? That can be done following the logic described in the programming section of the reference guide, but I agree it is time to make that part of the core packages.

[Doxdrum]

Excellent suggestion! I'll sent you another merge request tomorrow... with the changes.

And, yes!, you're right about the operator D. Thank you!