I am a computer science PhD student, and I need statistical manifolds theory for my work. I am currently reading Differential Geometry of Curves and Surface by Kristopher Tapp and Carmo. I plan to study Lee's smooth Manifold next. (My supervisor's recommendations.)
Unfortunately I do not have the topology background to the depth I would like to have to; I studied Topology without Tears in my second year (not completely) but I do not remember all of what I read either. But I am keeping it alongside.
Can anyone suggest a self-readable set of books to get to Stat. Manifold? I might end up needing quite a bit of Discrete Differential Geometry as well. Other suggestions about alternate study plans are also welcome.
You can try Information Geometry and Its Applications, by Shun-ichi Amari. He pretty much founded the field, and in this book he doesn't really assume any background on smooth manifolds.
Another option is Information Geometry, by Ay, Jost, Le, and Schwachhofer, but the mathematics there is much more rigorous than in Amari's book.
For something quicker, look up An Elementary Introduction to Information Geometry: it is a 61-page review by Frank Nielsen, published on 'entropy'.