Hello, 
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the pullback of the Thom class by the zero section, but I would like a proof of the fact that this class gives the Euler characteristic when coupled to the fundamental class. Thanks in advance. 



Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?
For a complete proof of the Gauss-Bonnet-Chern for *arbitrary* vector bundles (not just tangent bundles) see Section 8.3.2 of [these notes](http://www.nd.edu/~lnicolae/Lectures.pdf). The proof is Chern's original proof, based on Chern-Weil theory, but the language is more modern.


For a purely topological proof, see Section 5.3 of [these notes](http://www.nd.edu/~lnicolae/MS.pdf).


