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I am currently doing an undergraduate project about Gauss-Bonnet-Chern Theorem. Is there any particular books/papers regarding the application of the theorem in the theory of general relativity?

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Gauss-Bonnet is used extensively in the proofs of “no-hair” and “positive mass”:

Israel, Werner, Event Horizons in Static Vacuum Space-Times, Phys. Rev. 164, 1776-1779 (1967).

Schoen, Richard; Yau, Shing-Tung, On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys. 65, 45-76 (1979). ZBL0405.53045.

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