Chern's conjecture for closed affine manifolds and its Euler–Satake analogue for compact affine orbifolds

An affine manifold is a manifold whose coordinate changes are affine transformations of model Euclidean space, equivalently a manifold carrying a flat, torsion-free connection on its tangent bundle. Around 1955, S. S. Chern conjectured that the Euler characteristic of every closed (compact, boundaryless) affine manifold vanishes. Bagaev and Zhukova (2019) formulated the orbifold analogue — the Euler–Satake characteristic of every compact affine orbifold vanishes — proved it under the orbifold versions of the classical sufficient conditions (completeness, existence of a parallel volume form), and showed that this orbifold statement is equivalent to the original manifold conjecture. The unrestricted problem — with no completeness, holonomy, or parallel-volume hypothesis — remains open in both formulations: apart from the two-dimensional case, every published result covers only a proper special class of structures, and no claimed general proof has survived review.

Background

An affine structure on an $n$-manifold $M$ is a maximal atlas whose transition functions are restrictions of maps $x \mapsto Ax+b$ with $A \in \mathrm{GL}(n,\mathbb R)$; equivalently, it is a flat (zero-curvature) torsion-free connection $\nabla$ on the tangent bundle $TM$, the connection in which the affine coordinates have vanishing Christoffel symbols. The torsion-free flat connection determines a holonomy representation $\pi_1(M) \to \mathrm{GL}(n,\mathbb R)$ together with a developing map from the universal cover to $\mathbb R^n$; the structure is called complete when the developing map is a diffeomorphism (so a complete compact affine manifold is a quotient $\mathbb R^n/\Gamma$, a compact flat Riemannian space form in the broader affine sense). The structure is called special affine when $M$ carries a $\nabla$-parallel volume form, i.e. the holonomy reduces to $\mathrm{SL}(n,\mathbb R)$, the subgroup of $\mathrm{GL}(n,\mathbb R)$ consisting of determinant-one matrices. Hopf manifolds $(\mathbb R^{n+1}\setminus\{0\})/\langle x \mapsto \lambda x\rangle$ are standard examples of closed affine manifolds that are not complete.

For a Riemannian metric the Chern–Gauss–Bonnet theorem expresses the Euler characteristic $\chi(M)$ as the integral of the Pfaffian of the curvature, so a flat Levi-Civita connection forces $\chi(M)=0$. The difficulty of Chern's problem is that a general flat torsion-free connection admits no compatible metric, so Chern–Weil theory cannot represent the Euler class, and indeed the Euler class of a flat vector bundle need not vanish. Around 1955 Chern conjectured that $\chi(M)=0$ for every closed affine manifold. The classical partial results are: the two-dimensional case (Benzécri, 1955); Milnor's 1958 inequalities, which characterize the oriented rank-two bundles over a surface that admit some flat connection and imply the surface case; the complete case (Kostant–Sullivan, 1975); and, much later, the special affine case (Klingler, Ann. of Math. 2017). Smillie (1977) showed that torsion-freeness is essential: for every even dimension $> 2$ there exist closed manifolds with nonzero Euler characteristic whose tangent bundle admits a flat connection that is not torsion-free. Further positive cases include manifolds locally a product of surfaces (Bucher–Gelander, 2011) and closed affinely flat manifolds whose holonomy action on $\mathbb{RP}^n$ preserves a probability measure (Jo–Kim, 2003); most recently, Liu (2025) confirmed the conjecture for compact Hessian manifolds — affine manifolds whose flat structure carries a metric that is locally the Hessian $D^2\varphi$ of a potential function. As the 2018 lecture notes of Arias Abad and Vélez Vásquez put it, the general case remains open, and survey literature through 2023 (Löh–Raptis) still treats it as open.

Orbifolds are the natural singular generalization of manifolds: spaces locally modeled on quotients $\mathbb R^n/G$ by finite groups of affine changes of coordinates, with compatible gluing. Satake (1957) extended Gauss–Bonnet to these $V$-manifolds: the Euler–Satake characteristic $\chi_{\mathrm{ES}}$ is the integral of the Euler form over the orbifold with each stratum weighted by the reciprocal of the order of its local isotropy group; equivalently it is the alternating cell sum with each cell weighted by $1/|\Gamma_{\text{cell}}|$. It agrees with the ordinary Euler characteristic on manifolds (trivial isotropy), is multiplicative under finite orbifold covers, and differs from both the topological Euler characteristic of the underlying space and from the stringy orbifold Euler characteristic of Chen–Ruan type (the invariant of orbifold cohomology theory in which the twisted sectors of the orbifold also contribute). Bagaev and Zhukova (2019) observed that since every compact orbifold admits a finite orbifold cover that is a manifold and $\chi_{\mathrm{ES}}$ is multiplicative, the conjecture 'if $M$ is a closed affine manifold then $\chi(M)=0$' is equivalent to its orbifold analogue 'if $\mathcal X$ is a compact affine orbifold then $\chi_{\mathrm{ES}}(\mathcal X)=0$'. They proved the orbifold analogue under orbifold versions of the Kostant–Sullivan (complete) and Klingler (special affine) hypotheses, and constructed examples showing that the direct analogues for the topological and the stringy orbifold Euler characteristics are false — the Euler–Satake characteristic is the invariant for which the conjectured vanishing can hold.

Problem Statement

Chern's conjecture (~1955) and its orbifold analogue: let $M$ be a closed (compact, without boundary) affine manifold, i.e. a manifold equipped with a flat torsion-free connection on its tangent bundle. Must the Euler characteristic $\chi(M)$ vanish? Equivalently — this is the formulation introduced by Bagaev and Zhukova (2019), equivalent to the manifold statement — must the Euler–Satake characteristic $\chi_{\mathrm{ES}}(\mathcal X)$ vanish for every compact affine orbifold $\mathcal X$ (an orbifold whose local orbifold charts are related by affine changes of coordinates)? Here $\chi_{\mathrm{ES}}$ is Satake's orbifold Euler characteristic: for a cell decomposition of the orbifold with each cell $\sigma$ stabilized by a finite local isotropy group $\Gamma_\sigma$, $\chi_{\mathrm{ES}}(\mathcal X) = \sum_{\sigma} (-1)^{\dim\sigma}/|\Gamma_\sigma|$, an invariant that agrees with the ordinary Euler characteristic when all isotropy is trivial and is multiplicative under finite orbifold covers. The question is to be answered without any additional hypothesis such as completeness of the affine structure, existence of a parallel volume form (special affine), local product structure, or an invariant measure for the holonomy: a complete proof covering all closed affine manifolds (all compact affine orbifolds), or an explicit counterexample — a closed affine manifold with $\chi(M) \neq 0$, equivalently a compact affine orbifold with $\chi_{\mathrm{ES}}(\mathcal X) \neq 0$ — would resolve it. By the equivalence, a resolution of the orbifold statement is simultaneously a resolution of the classical manifold conjecture, and vice versa.

References

1. Andrey V. Bagaev and Nina 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
2. Bruno Klingler, Chern's conjecture for special affine manifolds, Annals of Mathematics (2) 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. Camilo Arias Abad and Sebastián Vélez Vásquez, Lectures on the Euler characteristic of affine manifolds, arXiv:1802.03624 (2018), https://arxiv.org/abs/1802.03624
4. Bertram Kostant and Dennis Sullivan, The Euler characteristic of an affine space form is zero, Bulletin of the American Mathematical Society 81 (1975), no. 5, 937–938. DOI: 10.1090/s0002-9904-1975-13896-1, https://doi.org/10.1090/s0002-9904-1975-13896-1
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
6. 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
7. 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, https://doi.org/10.1016/j.aim.2011.06.022
8. Kyeonghee Jo and Hyuk Kim, Invariant measure and the Euler characteristic of projectively flat manifolds, Journal of the Korean Mathematical Society 40 (2003), no. 1, 109–128. DOI: 10.4134/jkms.2003.40.1.109, https://doi.org/10.4134/jkms.2003.40.1.109 (preprint: arXiv:math/0210335, https://arxiv.org/abs/math/0210335)
9. Ichirō Satake, The Gauss–Bonnet theorem for V-manifolds, Journal of the Mathematical Society of Japan 9 (1957), 464–492. DOI: 10.2969/jmsj/00940464, https://doi.org/10.2969/jmsj/00940464
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16. Clara Löh and George Raptis, A roadmap to the (vanishing of the) Euler characteristic, arXiv:2306.16933 (2023), https://arxiv.org/abs/2306.16933