Literature on Chern-Weil Theory and the Chern-Gauß-Bonnet Theorem At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and modern literature on this subject. Classically I know of Milnor's Appendix C in Milnor, Stasheff: "Characteristic Classes" and the famous text from Spivak's A Comprehensive Introduction to Differential Geometry entitled "The Generalized Gauß-Bonnet Theorem and What It Means To Mankind". Is there any more recent recommendable source?
During my search I also came along Zhang: Lectures on Chern-Weil Theory and Witten Deformations. Does anyone have practical experience with this book?
Thanks in advance!
 A: For reference's sake, here is Chern's original paper:
http://www.maths.ed.ac.uk/~aar/papers/chern7.pdf
The "modern" version of this theorem can be found in Liviu I. Nicolaescu's personal website, Lectures on the Geometry of Manifolds, Chapter 8. I have not read his proof, so cannot comment on it. Milnor&Stasheff's bilbliography listed a few other papers related to Chern class and Gauss-Bonnet theorem, which you can look up. 

One should note that the notations in Chern's paper is slightly different from the ones used in contemporary Riemannian geometry textbooks. You need to check Chern's Lectures on Differential geometry as a reference . 
For the most trivial example, he define Christoffel symbols by
$$
\nabla s_{\alpha}=\Gamma^{\beta}_{\alpha i}du_{i}\otimes s_{\beta}, \omega^{\beta}_{\alpha}=\Gamma^{\beta}_{\alpha i}du_{i}
$$
so we have
$$
\nabla s_{\alpha}=w^{\beta}_{\alpha}\otimes s_{\beta}
$$
and if one use $S=[s_{1}\cdots s_{n}]^{T}$, $\omega=\{\omega_{ij}\}$, then we can abbreviate this as
$$
\nabla S=\omega\otimes S
$$
and it only takes one line from his paper because he assumed that reader knows about it. However, I do not think this is explained very clearly in "modern style" textbooks (Jost, Lee, Taubes, etc). And this might make the reader reading his paper difficult. 
A: This book was not available four years ago but I thought I should include this here for posterity. There's this book by Loring W. Tu published recently in 2017 titled Differential Geometry: Connections, Curvature, and Characteristic Classes. It provides a really accessible introduction to Chern-Weil Theory, though the Chern-Gauß-Bonnet Theorem was only stated without proof.
