Some good books on application of Gauss-Bonnet-Chern Theorem in general relativity

Question

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

Edit: I should ask more specifically. Is there any good reference on the application of Gauss-Bonnet-Chern Theorem for four-dimensional manifold on general relativity?

Answer 1

I doubt you will have entire books devoted to the subject, as you hoped for.

Here are however two papers in which some version of the Gauss-Bonnet theorem (unfortunately not the 4 dimensional version) makes interesting appearances.

1. In a work of Galloway and Schoen, they studied the question of "what can be the topology of a black hole" extending on previous works of Hawking. Through analytic methods they conclude that the surface of the black hole must be a manifold that supports a metric of positive scalar curvature, which in the case of 4 dimensional spacetimes (where the black hole surface is two dimensional) tells you that the surface have to be spherical.

2. In a never published note of mine, I asked the question of whether there is an analogue to the positive mass theorem when looking only at spacetimes with 2 spatial and 1 time dimensions. (This was not entirely idle in view of holography and AdS-CFT correspondences.) In any case, the result follows from an application of the classical Gauss-Bonnet theorem.