On Chern's conjecture about the Euler characteristic of affine manifolds (2018)
The development the theory of characteristic classes allowed Shiing-Shen Chern to generalize the Gauss Bonnet theorem to Riemannian manifolds of arbitrary dimension. The Chern Gauss Bonnet theorem expresses the Euler characteristic as an integral of a polynomial evaluated on the curvature tensor, i.e if K is the curvature form of the Levi-Civita connection, the Chern Gauss Bonnet formula is . In particular, the theorem implies that if the Levi Civita connection is _at, the Euler characteristic is zero.An a_ne structure on a manifold is an atlas whose transition functions are a_ne transformations. The existence of such a structure is equivalent to the existence of a _at torsion free connection on the tangent bundle. Around 1955 Chern conjectured the following: Conjecture. The Euler characteristic of a closed affine manifold is zero. Not all fat torsion free connections on TM admit a compatible metric, and therefore, Chern-Weil theory cannot be used in general to write down the Euler class in terms of the curvature. In 1955, Benzécri [1] proved that a closed affine surface has zero Euler characteristic. Later, in 1958, Milnor [11] proved inequalities which completely characterise those oriented rank two bundles over a surface that admit a fiat connection. These inequalities prove that in case of a surface the condition "be torsion free" in Chern's conjecture is not necessary. In 1975, Kostant and Sullivan [9] proved Chern's conjecture in the case where the manifold is complete. In 1977, Smillie [15] proved that the condition that the connection is torsion free matters. For each even dimension greater than 2, Smillie constructed closed manifolds with non-zero Euler characteristic that admit a _at connection on their tangent bundle. In 2015, Klingler [14] proved the conjecture for special affine manifolds. That is, affine manifolds that admit a parallel volume form.