A conjecture attributed to S.-S. Chern (c. 1955) states that every closed affine manifold - a closed manifold whose tangent bundle carries a torsion-free flat connection, equivalently a manifold with an atlas whose transition maps are affine - has vanishing Euler characteristic. The conjecture is open in general: it is known for surfaces (via Milnor's 1958 inequality on flat rank-two bundles), for complete affine manifolds (Kostant-Sullivan 1975), and for special affine manifolds admitting a parallel volume form (Klingler, Annals of Mathematics 2017), while Smillie's 1977 examples of flat connections with torsion and nonzero Euler characteristic show the torsion-free hypothesis is essential. Klatt (2021) proved that for any connection with a fibre metric the Euler class is represented by Pf(Ω−14(g−1∇g)2)\mathrm{Pf}(\Omega - \tfrac{1}{4}(g^{-1}\nabla g)^2); for a flat connection this reduces the conjecture to the exactness of the Pfaffian correction form Pf((g−1∇g)2)\mathrm{Pf}((g^{-1}\nabla g)^2), whose cohomology class is independent of the auxiliary metric, and Klatt proposed (without pursuing) variational strategies for controlling that form. Claimed general proofs (Ge 2020, withdrawn; Cocos 2015-2025, unpublished preprints) have not been accepted, and 2025-2026 work confirms the conjecture only for further subclasses (compact Hessian manifolds - those whose metric is locally the Hessian of a potential in affine coordinates - Liu 2025). The general question remains open.

Background

An affine manifold is a manifold MM equipped with an atlas whose coordinate-transition maps lie in the affine group Aff(Rn)=GL(n,R)⋉Rn\mathrm{Aff}(\mathbb{R}^n)=\mathrm{GL}(n,\mathbb{R})\ltimes\mathbb{R}^n; equivalently, the tangent bundle of MM carries a torsion-free flat connection ∇\nabla (flat means the curvature Ω=0\Omega=0; torsion-free means ∇XY−∇YX−[X,Y]=0\nabla_X Y-\nabla_Y X-[X,Y]=0). Flat Riemannian manifolds are the basic examples, but many affine structures are incomplete: the developing map - the map from the universal cover to Rn\mathbb{R}^n obtained by analytically continuing affine charts along paths - need not be onto, so the manifold need not be a quotient of Euclidean space by an affine group action.

The Euler characteristic of a closed oriented manifold of dimension 2k2k is the evaluation of the Euler class e(TM)e(TM) - the characteristic class of the tangent bundle that obstructs the existence of a nowhere-vanishing vector field - on the fundamental class. The Chern-Gauss-Bonnet theorem computes it by Chern-Weil theory, which represents characteristic classes by invariant polynomials in the curvature of a connection, as χ(M)=∫MPf(Ωg/2π)\chi(M)=\int_M \mathrm{Pf}(\Omega^{g}/2\pi), where Ωg\Omega^{g} is the curvature of any connection compatible with a Riemannian metric gg and Pf\mathrm{Pf} is the Pfaffian, the polynomial on skew-symmetric 2k×2k2k\times 2k matrices satisfying Pf(A)2=det⁡A\mathrm{Pf}(A)^2=\det A. This works because the Pfaffian is invariant under conjugation by SO(2k)\mathrm{SO}(2k). For a general GL(2k,R)\mathrm{GL}(2k,\mathbb{R})-connection the curvature matrix is not skew-symmetric and the Pfaffian is not invariant under the full general linear group, so the Euler class is not a Chern-Weil class of an arbitrary connection. This is exactly why flatness does not formally force χ(M)=0\chi(M)=0: for invariant polynomials on gl(n,R)\mathfrak{gl}(n,\mathbb{R}) (for instance the Pontryagin classes) Chern-Weil does apply, so all Pontryagin classes of an affine manifold vanish rationally, but the Euler class escapes that argument and is the last rational characteristic-class obstruction.

Around 1955 S.-S. Chern conjectured that nevertheless χ(M)=0\chi(M)=0 for every closed affine manifold. The conjecture is confirmed only in restricted classes: closed affine surfaces (via Milnor's 1958 theorem that an oriented rank-two bundle EE over a closed oriented surface of genus gg admits a flat connection if and only if ∣D(E)∣<g|D(E)|<g, where D(E)=∫Σe(E)D(E)=\int_\Sigma e(E), which permits a flat tangent bundle only on the torus among closed surfaces); complete affine manifolds (Kostant-Sullivan 1975); and special affine manifolds, those carrying a parallel volume form, meaning a nowhere-vanishing volume form ω\omega with ∇ω=0\nabla\omega=0 (Klingler, Annals of Mathematics 2017, via a canonical para-hypercomplex structure on TMTM - a complex structure anti-commuting with a product structure - and spectral sequences). Conversely, Smillie (1977) constructed closed manifolds of every even dimension greater than two whose tangent bundles admit flat connections with torsion and nonzero Euler characteristic, so torsion-freeness is indispensable and any correct proof must use it. The survey lectures of Arias Abad and Vélez Vásquez (2018) record that the general case remains open, and preprints of 2025-2026 continue to treat it as open.

Klatt (2021) proved a formula that expresses the conjecture in connection-theoretic terms: for any connection ∇\nabla with curvature Ω\Omega on an oriented rank-2k2k real vector bundle EE with fibre metric gg,

(2π)ke(E)=Pf(Ω−14(g−1∇g)2),(2\pi)^k e(E)=\mathrm{Pf}\left(\Omega-\tfrac{1}{4}(g^{-1}\nabla g)^2\right),

where g−1∇gg^{-1}\nabla g is the End(E)\mathrm{End}(E)-valued 1-form obtained by differentiating the metric with the connection (one index of ∇g\nabla g raised by gg) and the square is the wedge product. When ∇\nabla is flat, the correction form Pf((g−1∇g)2)\mathrm{Pf}((g^{-1}\nabla g)^2) is closed and represents the fixed class (−4)k(2π)ke(E)(-4)^k(2\pi)^k e(E) for every metric gg. Applied to E=TME=TM of a closed 2k2k-manifold with a torsion-free flat connection, Chern's conjecture is therefore equivalent to exactness of Pf((g−1∇g)2)\mathrm{Pf}((g^{-1}\nabla g)^2) for one - and hence every - Riemannian metric gg. Klatt proposed to seek metrics adapted to ∇\nabla, for example minimizers of the scale-invariant functional F∇(g)=∫M∣∇g∣2k dVg\mathcal{F}_{\nabla}(g)=\int_M |\nabla g|^{2k}\,\mathrm{d}V_g or of the volume-normalized functional K∇(g)=∫M∣∇g∣2 dVg\mathcal{K}_{\nabla}(g)=\int_M |\nabla g|^{2}\,\mathrm{d}V_g, but left the existence, regularity, and topological consequences of such minimizers open.

Problem Statement

Chern's conjecture (c. 1955), in the equivalent Euler-form reformulation of Klatt (2021). Let MM be a closed (compact, without boundary) manifold of even dimension 2k2k whose tangent bundle admits a torsion-free flat connection ∇\nabla - a closed affine manifold. Prove that χ(M)=0\chi(M)=0, or exhibit a closed affine manifold with χ(M)≠0\chi(M)\ne 0. Closed odd-dimensional manifolds have χ=0\chi=0 automatically by Poincaré duality, so the even-dimensional statement is the full conjecture; non-orientable cases reduce to orientable ones by passing to the orientation double cover, which inherits an affine structure and has Euler characteristic 2χ(M)2\chi(M).

Equivalently, when MM is oriented: for such a connection ∇\nabla and any Riemannian metric gg on MM, the closed 2k2k-form Pf((g−1∇g)2)\mathrm{Pf}((g^{-1}\nabla g)^2) represents (−4)k(2π)ke(TM)(-4)^k(2\pi)^k e(TM) in de Rham cohomology, and the conjecture asserts that this form is exact - for a single metric gg if and only if for every metric gg, since its cohomology class does not depend on gg.

The known cases (dimension two; complete affine manifolds; special affine manifolds with parallel volume; compact Hessian manifolds) do not settle the general case.

References

  1. Brian Klatt, The Euler Class from a General Connection, Relative to a Metric, arXiv:2101.06791 (2021), DOI 10.48550/arXiv.2101.06791, https://arxiv.org/abs/2101.06791 (source paper of the Euler-form reformulation)
  2. Bruno Klingler, Chern's conjecture for special affine manifolds, Annals of Mathematics 186 (2017), no. 1, 69-95, DOI 10.4007/annals.2017.186.1.2, https://doi.org/10.4007/annals.2017.186.1.2
  3. B. Kostant and D. Sullivan, The Euler characteristic of an affine space form is zero, Bulletin of the American Mathematical Society 81 (1975), 937-938, DOI 10.1090/S0002-9904-1975-13896-1, https://doi.org/10.1090/S0002-9904-1975-13896-1
  4. John Milnor, On the existence of a connection with curvature zero, Commentarii Mathematici Helvetici 32 (1958), 215-223, DOI 10.1007/BF02564579, https://doi.org/10.1007/BF02564579
  5. John Smillie, Flat manifolds with non-zero Euler characteristics, Commentarii Mathematici Helvetici 52 (1977), 453-455, DOI 10.1007/BF02567378, https://doi.org/10.1007/BF02567378
  6. Camilo Arias Abad and Sebastián Vélez Vásquez, Lectures on the Euler characteristic of affine manifolds, arXiv:1802.03624 (2018), DOI 10.48550/arXiv.1802.03624, https://arxiv.org/abs/1802.03624
  7. Jianquan Ge, Proof of Chern's conjecture on affine manifolds, arXiv:2002.03105 (2020, withdrawn by the author after errors were found), https://arxiv.org/abs/2002.03105
  8. Mihail Cocos, Proof of Chern conjecture for flat affine manifolds, arXiv:1504.04852 (2015, preprint, version 12 updated 2025, unpublished), https://arxiv.org/abs/1504.04852
  9. Mihail Cocos, On the topology of compact affine manifolds, arXiv:math/0511303 (2005, last revised 2012, marked Incomplete on arXiv), https://arxiv.org/abs/math/0511303
  10. Mihail Cocos, Affine-Orthogonal Manifolds and Deformation to Levi-Civita Connections, arXiv:2510.09648 (2025), https://arxiv.org/abs/2510.09648
  11. Hanwen Liu, On Topology of Compact Hessian Manifolds, arXiv:2509.01176 (2025), https://arxiv.org/abs/2509.01176
  12. Brian Klatt, Compact 4-Dimensional Spin Gradient m-quasi-Einstein Manifolds Satisfy the Hitchin-Thorpe Inequality when m>=1, arXiv:2012.13848 (2020), https://arxiv.org/abs/2012.13848
  13. Brian Klatt, A Generalization of Thorpe's Inequality, arXiv:2106.13848 (2021), https://arxiv.org/abs/2106.13848
  14. Yong Wang, Affine connections and Gauss-Bonnet theorems in the Heisenberg group, arXiv:2102.01907 (2021), https://arxiv.org/abs/2102.01907
  15. Tong Wu and Yong Wang, Gauss-Bonnet theorems associated to deformed Schouten-Van Kampen connection in the affine group and the group of rigid motions of the Minkowski plane, arXiv:2108.13595 (2021), https://arxiv.org/abs/2108.13595