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.
Once the library is publicaly available, a link will be provided.