# Reference request-What is the prerequisite of S.S.Chern's proof of the generalised Gauss-Bonnet theorem?

The title basically explains everything. The OP is an independent learner, who in the current stage sets S.S.Chern's proof of the generalised Gauss-Bonnet theorem as the goal. But what is the prerequisite knowledge of reading his two papers? I'm afraid basic Riemannian geometry would not suffice.

• Chern's proof of the original Gauss-Bonnet theorem is very pleasant and conceptual. It only requires manifold theory to the point of Stoke's theorem. There's a pretty good version of it in Berger's Panoramic View of Riemannian Geometry. I have a version written up in my differential geometry lecture notes, available on my webpage. The proof gives you a very good idea of what to expect in higher dimensions, and it's pleasantly conceptual. May 19, 2014 at 0:52
• another user [Rolf W. Walter] commented: There is a proof of the generalized Gauss/Bonnet theorem without characteristic classes etc in the book of mine `Walter, Rolf: Differential Geometry', Chapter 4, in German language. Jun 1, 2015 at 20:55

Here's a (possibly incomplete list) of some good things to know:

• Riemannian geometry, including a good understanding of curvature.

• Some algebraic topology, including familiarity with differential forms and De Rham cohomology. I first learned homology from Hatcher's Algebraic Topology and de Rham cohomology from Bott and Tu's Differential Forms in Algebraic Topology, but there are many other sources.

• Vector bundles, and Chern-Weil theory of characteristic classes. Milnor and Stasheff's Characteristic Classes is a classic; there's an appendix on Chern-Weil theory, which may be enough to get you started.

Edti: The idea of characteristic classes is to study a vector bundle over a manifold $M$ by associating to the vector bundle various distinguished cohomology classes in the cohomology of $M$. Loosely, these classes are supposed to measure how "twisted" (or "nontrivial") the vector bundle is. One can take a purely topological approach to this subject, but Chern-Weil theory is a more geometric approach, in which the cohomology classes are represented by explicit differential forms that are constructed from the curvature of a connection on the vector bundle. I think one could make the following analogy: just as C-G-B connects the topology and geometry of a manifold, Chern-Weil theory connects the topology and geometry of vector bundles.

Perhaps I was wrong to include characteristic classes in the list above. Strictly speaking, you actually don't need to study characteristic classes (or even vector bundles) in general to understand the C-G-B theorem, but might be good to do so to gain a little perspective. Maybe you can start attacking the proof of C-G-B, and come back to them as necessary.

The $n$-form (for $n$-dimensional $M$) that appears as the integrand in the C-G-B theorem is precisely the differential form representing the Euler class of the tangent bundle $TM$ that one obtains via Chern-Weil theory. For surfaces, the form is $K \, dA$; for higher-dimensional $M$, the form is a more complicated curvature expression, but it is always obtained from a polynomial in the curvature called the Pfaffian.

Searching mathoverflow turned up a few relevant questions and answers (among others, there's this one, which has links to several good sources).

Finally (probably in the long term, after you've read some of the above), if you are interested in analysis and PDE, you may appreciate learning some index theory, which involves beautiful connections between analysis, geometry, and topology. The Atiyah-Singer index theorem is a broad generalization of Chern-Gauss-Bonnet; you can get a quick introduction on Wikipedia. I got an introduction to this field from John Roe's Elliptic Operators, Topology, and Asymptotic Methods.

• Thank you for answering with such enthusiasm, but can you please tell me what Chern-Weil theory is? I don't seem to have had an understanding from normal internet materials May 19, 2014 at 10:29
• I edited my answer with a few more comments on Chern-Weil theory and characteristic classes. Hope it's helpful. Don't worry if my answer doesn't make sense right now; just keep reading and asking questions! (Besides, my answer probably isn't too coherent anyway.) May 20, 2014 at 2:33
• It seems hard to get Pfaffian from Chern-Weil theory? Isn't Chern-Weil assume the bundle tobe complex vector bundle, and I think the tangent bundle of arbitrary real manifold won't admit a complex vector bundle structure.
– lee
May 20, 2014 at 2:53
• I only know that the Chern-Weil theory will give the G-B-C theorem for complex manifolds. The way I know to get Pfaffian is that from Mathai-Quillen's geometric construction of Thom classes.
– lee
May 20, 2014 at 2:55
• Oh, BTW, the 2-dimensional case can be proved by Chern-Weil theory, as it is a well-known fact that all 2-real-dim'l Riemannian manifolds are Riemann surfaces.
– lee
May 20, 2014 at 2:57