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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?

stf91
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  • manifolds is a branch, almost. – Bhaskar Vashishth Jan 06 '20 at 16:21
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    differential geometry – Bhaskar Vashishth Jan 06 '20 at 16:22
  • Differential geometry introduces the basic concepts. The specific topic for general relativity should be Lorentzian manifolds. –  Jan 06 '20 at 16:22
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    Differential geometry itself is part of a broader background topic called differential topology, so if you found differential geometry heavy going you could also start with a more elementary book on differential topology. And then, of course, you could go back even further and start with a book on (general) topology. – Lee Mosher Jan 06 '20 at 16:44
  • @Lee: that 's very useful information, thanks. – stf91 Jan 06 '20 at 17:12

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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.

oyster
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  • The author specifically refers to mathematics graduate students with some knowledge of global differential geometry. What does the word global mean here? – stf91 Jan 06 '20 at 16:50
  • My guess is that they are Riemmanian: those can include any kind of geometry. They don't even require a 'fixed' metric like in spherical or hyperbolic geometry. For GR use Riemmanian geometry, for special relativity use the lorentz metric (-1,1,1,1) – oyster Jan 06 '20 at 16:59