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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?

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Where exactly in the book is this outline? – Jonas Meyer Jan 11 '12 at 0:36
@Jonas: I've added the location to the question. – Jon Bannon Jan 16 '12 at 13:07

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.

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Thank you for the answer, Iyengar! – Jon Bannon Mar 20 '12 at 13:29
Thanks a lot sir @JonBannon – Iyengar Mar 20 '12 at 16:48
@JonBannon : But, can I ask you one thing sir ? . In the article you have provided, I can't find any proof of Gauss-Bonnet Theorem, but it simply uses some remarkable stability which is a consequence of that theorem and by combining with the theory of multi-linear forms , one can show as an analogue that some trilinear map should be some constant as predicted by the Gauss-Bonnet theorem. But Am I right ? . – Iyengar Mar 20 '12 at 17:07

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