An Introduction to the Chern Conjecture

The aim of the thesis is to give an introduction to the Chern conjecture: a long-standing conjecture which claims that closed affine manifolds have zero Euler characteristic. We also give an exposition of the necessary background about characteristic classes, with a particular focus on Chern-Weil theory. This is done by presenting some of the main (partial) results about the conjecture. We first study affine manifolds and introduce the Euler class with its topological definition. Later we focus on characteristic classes and classifying spaces. We present some classical results by Benzécri, Milnor and Smillie, which solve the conjecture in dimension 2 and show that, in general dimensions, the hypotheses of the conjecture can not be weakened. We finally focus on Chern-Weil theory and then present Kostant and Sullivan's proof of the conjecture in the case of complete affine manifolds