I wonder what characteristics of Riemannian manifold can be expressed in terms of metric? Are there any results similar to Gauss–Bonnet theorem? Does the Riemannian metric give any information about betti numbers, homotopy groups and other characteristics? Thanks for helping.

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    $\begingroup$ This is way too broad, did you try to google "curvature and homology"? Or Cartan-Hadamard theorem? What textbook are you reading? $\endgroup$ – Moishe Kohan Aug 24 '14 at 23:20
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    $\begingroup$ Or Chern-Gauss-Bonnet ... Look at Bishop-Goldberg, for example. $\endgroup$ – Ted Shifrin Aug 24 '14 at 23:25

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