What branch of mathematics studies manifolds?

Question

Hi: I am considering to study a book called General Relativity for Mathematicians. In the book the word manifold appears many times. Consider manifolds in the context of the general relativity. What could be the branch of mathematics studying them? In google I see there is a thing called topological manifold. Is it geometry? Is it topology? Is it mathematical analysis? Is it differential geometry?

Answer 1

If you want to study the mathematics behind GR I recommend that you go to newlecturesGR

I found it to be a very good start, if you want to start with the mathematics of it grab a book on Riemmanian manifolds.