Back in graduate school whenever I needed to do some local computations on a manifold, I found that I had to program the method from scratch. Over the years, I ended up with a bunch of disparate pieces. Many years later, I decided to merge those pieces into a coherent library for sage.
Currently I am in the final stages of getting the initial release ready to go.
I am aware that in the intervening years SageManifolds package became available. However, from what I saw when I looked at it, it seems a much bigger library, focused on manifolds as a whole, whereas my library focuses on doing everything in local coordinates.
This package is now available on github. The package is by no means perfect, but it works.
Documentation is complete, while I would prefer to write more tests, the current tests provide a nice foundation by testing a wide variety of calculations on the standard examples.