Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem.
Chern's conjecture states that the Euler characteristic of a compact affine manifold vanishes.
In case the connection ∇ is the Levi-Civita connection of a Riemannian metric, the Chern–Gauss–Bonnet formula:
implies that the Euler characteristic is zero. However, not all flat torsion-free connections on 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.
The conjecture is known to hold in several special cases:
Additionally obtained related results:
For flat pseudo-Riemannian manifolds or complex affine manifolds, this follows from the Chern–Gauss–Bonnet theorem.
Also, as proven by M.W. Hirsch and William Thurston in 1975 for incomplete affine manifolds, the conjecture holds if the holonomy group is a finite extension, a free product of amenable groups (however, their result applies to any flat bundles over manifolds). [3]
In 1977, John Smillie produced a manifold with the tangent bundle with nonzero-torsion flat connection and nonzero Euler characteristic, thus he disproved the strong version of the conjecture asking whether the Euler characteristic of a closed flat manifold vanishes. [2]
Later, Huyk Kim and Hyunkoo Lee proved for affine manifolds, and more generally projective manifolds developing into anaffine space with amenable holonomy by a different technique using nonstandard polyhedral Gauss–Bonnet theorem developed by Ethan Bloch and Kim and Lee. [4] [5]
In 2002, Suhyoung Choi slightly generalized the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is 0. He showed that if an even-dimensional manifold is obtained from a connected sum operation from K(π, 1)s with amenable fundamental groups, then the manifold does not admit an affine structure (generalizing a result of Smillie). [6]
In 2008, after Smillie's simple examples of closed manifolds with flat tangent bundles (these would have affine connections with zero curvature, but possibly nonzero torsion), Bucher and Gelander obtained further results in this direction.
In 2015, Mihail Cocos proposed a possible way to solve the conjecture and proved that the Euler characteristic of a closed even-dimensional affine manifold vanishes.
In 2016, Huitao Feng (Chinese :冯惠涛) and Weiping Zhang, both of Nankai University, claimed to prove the conjecture in general case, but a serious flaw had been found, so the claim was thereafter retracted. After the correction, their current result is a formula that counts the Euler number of a flat vector bundle in terms of vertices of transversal open coverings. [7]
Notoriously, the intrinsic Chern–Gauss–Bonnet theorem proved by Chern that the Euler characteristic of a closed affine manifold is 0 applies only to orthogonal connections, not linear ones, hence why the conjecture remains open in this generality (affine manifolds are considerably more complicated than Riemannian manifolds, where metric completeness is equivalent to geodesic completeness).
There also exists a related conjecture by Mikhail Leonidovich Gromov on the vanishing of bounded cohomology of affine manifolds. [8]
The conjecture of Chern can be considered a particular case of the following conjecture:
A closed aspherical manifold with nonzero Euler characteristic doesn't admit a flat structure
This conjecture was originally stated for general closed manifolds, not just for aspherical ones (but due to Smillie, there's a counterexample), and it itself can, in turn, also be considered a special case of even more general conjecture:
A closed aspherical manifold with nonzero simplicial volume doesn't admit a flat structure
While generalizing the Chern's conjecture on affine manifolds in these ways, it's known as the generalized Chern conjecture for manifolds that are locally a product of surfaces.
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