In Connes’s book on noncommutative geometry, he outlines a rather short “algebraic” proof of the Gauss-Bonnet theorem that uses multilinear forms. (Start reading on page 19 of the book) This is given as motivation for cyclic cohomology.
Where can I find a fleshed-out version of this proof?
After many days of consistent effort, finally I could find a complete article that speaks about proving the various corollaries and theorems of Gauss-Bonnet , even though the article starts with a preliminary version of the proof ( considering Riemann metrics and the Euler forms, it does have a different versions of the proof, and the whole article is related to that ).
It does have a usage of Multi-linear forms ( in the chapter ‘curvature’ ) , but I think this is an elegant article that completely speaks about the Gauss-Bonnet Theorem in various view points. The article which I am talking about is Lectures on geometry of manifolds by Liviu I. Nicolaescu .
Thank you. I think it will surely serve your purpose. I will edit and add some more articles, once I verify that they contain something related to this stuff.