Formulation of the open question extracted from M. Cocos, 'A note on locally metric connections' (arXiv:1606.05813): Chern's conjecture (c. 1955) asserts that every closed affinely flat manifold — one whose tangent bundle carries a flat, torsion-free linear connection, equivalently one with an atlas whose coordinate changes are affine maps — has vanishing Euler characteristic. The conjecture remains open in the accepted literature. It is confirmed only under auxiliary hypotheses: complete affine manifolds (Kostant–Sullivan 1975), surfaces (Milnor 1958), special affine manifolds with parallel volume (Klingler 2017), manifolds locally a product of surfaces (Bucher–Gelander 2011), aspherical manifolds with injective holonomy containing a pure translation (Bucher–Connell–Lafont 2018), and compact Hessian manifolds (Liu 2025); for compact affine orbifolds, Bagaev–Zhukova (2019) proved the Euler–Satake analogue under the corresponding sufficient conditions and showed that its general form is equivalent to the manifold conjecture. An unrefereed preprint series by the source paper's author (arXiv:1504.04852, 2015–2025) claims a full proof but is unpublished and unacknowledged by the community, so it does not alter the status. The source paper's derived route — path-connectedness of the space of locally metric connections implying the conjecture — is likewise open.

Background

An affine (or affinely flat) manifold is a smooth manifold MM whose tangent bundle carries a linear connection ∇\nabla that is flat (its curvature vanishes identically) and torsion-free (∇XY−∇YX=[X,Y]\nabla_X Y-\nabla_Y X=[X,Y]). Equivalently, MM admits an atlas whose transition functions are restrictions of affine maps x↦Ax+bx\mapsto Ax+b of Rn\mathbb{R}^n; the linear parts of the transitions around loops form the holonomy group, the group of linear parts of parallel transport. Examples include flat Riemannian manifolds and quotients of convex domains, but also incomplete exotic quotients. The Euler characteristic χ(M)\chi(M) is the alternating sum of the Betti numbers; it vanishes automatically on closed odd-dimensional manifolds, so the conjecture below concerns even dimensions.

Chern's conjecture (c. 1955) states that the Euler characteristic of a closed (compact, without boundary) affine manifold is zero. The natural proof strategy fails at a specific, well-understood point: the Chern–Gauss–Bonnet theorem computes χ(M)\chi(M) by integrating the Pfaffian of the curvature of a metric connection (one preserving a bundle metric), so a globally flat Riemannian metric gives χ=0\chi=0 immediately; but the Euler form (the Pfaffian curvature form representing the Euler class) is simply not defined for a general non-metric connection. Chern–Weil theory applied to the flat connection does force the rational Pontryagin classes of TMTM to vanish, yet it leaves the Euler class uncontrolled — Milnor showed in 1958 that flat oriented rank-2 bundles over a surface can have any Euler number within the Milnor–Wood range ∣e∣≤g−1|e|\le g-1, and used this to settle the conjecture for surfaces. The torsion-free hypothesis is essential: Smillie (1977) constructed, in every even dimension greater than 2, closed manifolds with nonzero Euler characteristic whose tangent bundles admit flat connections with torsion.

The conjecture is confirmed in several classes, each requiring an additional hypothesis: complete affine manifolds (Kostant–Sullivan 1975, where the universal cover develops diffeomorphically onto Rn\mathbb{R}^n); special affine manifolds, i.e. those admitting a parallel volume form (a nowhere-vanishing top-degree form parallel under ∇\nabla, equivalently linear holonomy inside SL(n,R)\mathrm{SL}(n,\mathbb{R}); Klingler 2017, Annals of Mathematics); closed manifolds locally a product of surfaces, via a sharp Milnor–Wood inequality for such metrics (Bucher–Gelander 2011); aspherical manifolds (those with contractible universal cover) whose affine holonomy is injective and contains a pure translation, via vanishing simplicial volume — the infimal total size of real singular simplices representing the fundamental class, whose vanishing forces χ=0\chi=0 in this setting (Bucher–Connell–Lafont 2018); compact Hessian manifolds, where the affine structure carries a metric that is locally the Hessian of a potential function (Liu 2025); and compact affine orbifolds for the Euler–Satake characteristic, the orbifold analogue of the Euler characteristic (Bagaev–Zhukova 2019, who prove the orbifold analogue is equivalent to the manifold conjecture). Goldman–Hirsch (1984) showed that higher-rank irreducible locally symmetric manifolds admit no affine structure at all. Lecture notes of Arias Abad–Vélez Vásquez (2018) summarize this history and state: 'As far as we can tell, the general case of Chern's conjecture remains open.'

The source paper (Cocos 2016, 'A note on locally metric connections') relates the conjecture to the theory of locally metric connections: a connection is locally metric if every point has a neighborhood on which some (possibly local) bundle metric is parallel with respect to it. The author proved earlier (Cocos 2010, Geometriae Dedicata) that if the space of locally metric connections on the tangent bundle of a compact affinely flat manifold is path-connected, then Chern's statement follows. This is a derived sufficient condition — a potential route, not a reformulation — and the source paper itself leaves both the conjecture and this sufficient condition unresolved, alongside two secondary open threads it raises: practical, frame-based criteria for deciding local metrizability beyond rank-2 bundles over surfaces, and the classification of metric-equivalence classes of locally metric connections (its Theorem 2.1 gives only the necessary condition that metric-equivalent connections have equal Euler class).

Problem Statement

Chern's conjecture (c. 1955): let MM be a closed (compact, without boundary) smooth manifold of even dimension n≥2n\ge 2 that is affinely flat, meaning the tangent bundle TMTM carries a flat, torsion-free linear connection ∇\nabla, or equivalently MM admits an atlas whose transition functions 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. Must the Euler characteristic χ(M)\chi(M) — the alternating sum ∑i(−1)idim⁡Hi(M;R)\sum_i (-1)^i \dim H_i(M;\mathbb{R}) of the Betti numbers of MM — vanish? Equivalently: is it the case that no closed manifold with χ(M)≠0\chi(M)\ne 0 admits an affine structure at all?

The conjecture is posed in full generality: no geodesic completeness, no parallel volume form, no asphericity, and, within the positive even dimensions n≥2n\ge 2 that the conjecture concerns, no restriction on topology or holonomy. Either outcome — a proof that χ(M)=0\chi(M)=0 for every closed affinely flat MM of even dimension n≥2n\ge 2, or a closed smooth manifold MM of dimension n≥2n\ge 2 with χ(M)≠0\chi(M)\ne 0 carrying a flat, torsion-free affine connection on TMTM — settles the question. The path-connectedness of the space of locally metric connections on TMTM of a compact affinely flat manifold, which Cocos (2010) proved sufficient for the conjecture, would not by itself resolve the conjecture.

References

  1. Mihail Cocos, 'A note on locally metric connections', arXiv:1606.05813 (2016), DOI 10.48550/arXiv.1606.05813, https://arxiv.org/abs/1606.05813 (source paper)
  2. Mihail Cocos, 'The deformation of flat connections and affine manifolds', Geometriae Dedicata 144 (2010), 71-78, DOI 10.1007/s10711-009-9387-9, https://doi.org/10.1007/s10711-009-9387-9
  3. Mihail Cocos, 'Proof of Chern conjecture for flat affine manifolds' (v1, April 2015, titled 'Pointwise metric compatible connections and a conjecture of Chern on affine manifolds'; v12, December 2025), arXiv:1504.04852, DOI 10.48550/arXiv.1504.04852, https://arxiv.org/abs/1504.04852
  4. Mihail Cocos, 'Affine-Orthogonal Manifolds and Deformation to Levi-Civita Connections', arXiv:2510.09648 (2025), DOI 10.48550/arXiv.2510.09648, https://arxiv.org/abs/2510.09648
  5. 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
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  7. 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
  8. William Goldman and Morris W. Hirsch, 'The radiance obstruction and parallel forms on affine manifolds', Transactions of the American Mathematical Society 286 (1984), 629-649, DOI 10.1090/S0002-9947-1984-0760977-7, https://doi.org/10.1090/S0002-9947-1984-0760977-7
  9. 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
  10. Michelle Bucher and Tsachik Gelander, 'The generalized Chern conjecture for manifolds that are locally a product of surfaces', Advances in Mathematics 228 (2011), no. 3, 1503-1542, DOI 10.1016/j.aim.2011.06.022, preprint arXiv:0902.1215, https://arxiv.org/abs/0902.1215
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  12. A. V. Bagaev and N. I. Zhukova, 'An analog of Chern's conjecture for the Euler-Satake characteristic of affine orbifolds', Journal of Geometry and Physics 142 (2019), 80-91, DOI 10.1016/j.geomphys.2019.04.002, https://doi.org/10.1016/j.geomphys.2019.04.002
  13. 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
  14. 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
  15. Hanwen Liu, 'On Topology of Compact Hessian Manifolds', arXiv:2509.01176 (2025), DOI 10.48550/arXiv.2509.01176, https://arxiv.org/abs/2509.01176