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I'm an undergraduate Math student and I am doing a side project for which I would like to study Differential geometry and its applications in general relativity. I have taken a few proof based courses including Analysis I, Introductory Number theory, Abstract Algebra.

I would like to get a text that isn't too far above what I have studied but still challenging and informative. Any suggestions are welcome.

Thanks

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    $\begingroup$ This isn't a serious recommendation since I haven't taken a peek at the text yet, but I hear that advanced undergraduates sometimes use Wald's GR. $\endgroup$ – nettle Sep 16 '15 at 20:59
  • $\begingroup$ doCarmo's pretty great in the Differential/Riemannian Geometry sector $\endgroup$ – S Valera Sep 16 '15 at 21:00
  • $\begingroup$ Winitzki has a book with topics on this material. While it's not comprehensive, it is freely available from the author. $\endgroup$ – pjs36 Sep 16 '15 at 21:02
  • $\begingroup$ This is worth looking at: cambridge.org/us/academic/subjects/physics/… $\endgroup$ – Simon S Sep 16 '15 at 21:02
  • $\begingroup$ do Carmo's does not deal with lorentzian geometry, though... S Valera. $\endgroup$ – CvZ Sep 16 '15 at 21:03
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Wald's General Relativity.

It is a book from a Physicist to Physicists, but any reader will not be missing statements and proofs (or sketch of them). His approach is still akin to formulations using coordinates, although he manages to do it in a global way.

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