TITLE: Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?
QUESTION [8 upvotes]: 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.

REPLY [10 votes]: 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. 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.