Changing chart multiple times in sagemanifolds

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Can we catch the KeyError and raise a more meaningful error message instead?

eric_g: Thanks for the answer. I assume it is correct, but it seems a surprising situation. Since $\psi_3\circ\psi_1^{-1} = (\psi_3\circ\psi_2^{-1})\circ(\psi_2\circ\psi_1^{-1})$, I don't see why the inverse maps should need to be defined. In particular, in the example I'm thinking of (converting standard Schwarzschild coordinates to Regge-Wheeler tortoise coordinates, then Eddington-Finkelstein null coordinates, and then Kruskal), it is impossible to invert the first coordinate transform. Does this mean that, while I can directly define a transition map from the first coordinate system to the third, I can't define it by going from the first to the second and then the second to the third? This seems like an unnecessary restriction.

rburing: I would also like clearer error message.

@my_screen_name: you are right: in the current stable version of Sage (8.8), there are unnecessary restrictions in the handling of coordinate changes. This is fixed by the Trac ticket #28072, which has been merged in Sage 8.9.beta2. In particular, with run with Sage 8.9.beta8, your original code gives no error and displays $g = \mathrm{d}z\otimes\mathrm{d}z$ in the last line. I have edited my answer accordingly.

@eric_g: Thanks for the further update, and I'm glad to learn this will be fixed in the next release.

SageMath 8.9 is out now. You can check that your example works nicely with it.