The Chern–Hopf sign conjecture: sign of the Euler characteristic of closed manifolds of nonpositive or negative sectional curvature

For a closed Riemannian manifold $M$ of even dimension $2n$, the classical conjecture attributed to H. Hopf (in print from S. S. Chern's 1955 paper, and listed as Problem 10 in Yau's 1982 problem list) predicts that nonpositive sectional curvature forces $(-1)^n\chi(M)\geq 0$, and strictly negative sectional curvature forces $(-1)^n\chi(M)>0$. The case $n\leq 2$ (dimensions 2 and 4) is classical, and the conjecture is confirmed under additional structure (Kähler metrics, curvature pinching, hard-Lefschetz-type conditions, small Nijenhuis tensor, symmetry), but the general case in dimensions $2n\geq 6$ with no extra hypotheses remains open.

Background

The Euler characteristic $\chi(M)$ of a closed (compact, without boundary) smooth manifold is the alternating sum of its Betti numbers. A Riemannian metric on $M$ assigns to each tangent 2-plane a sectional curvature $K$, the Gaussian curvature of the surface swept out by geodesics tangent to that plane; the conditions $K\leq 0$ or $K<0$ mean this holds for every 2-plane at every point. The Gauss–Bonnet–Chern theorem expresses $\chi(M)$ as the integral of a universal polynomial (the Pfaffian) in the curvature tensor, so the question of how the sign of curvature constrains $\chi(M)$ is one of the oldest problems in Riemannian geometry: it was asked by H. Hopf in the 1930s, appeared in print in S. S. Chern's 1955 Hamburg paper (p. 123), which also settled dimension 4 crediting the observation to J. Milnor, and was popularized as Problem 10 of S.-T. Yau's 1982 problem list.

What is known: in dimension 2 the statement is exactly the classical Gauss–Bonnet theorem, and in dimension 4 the Gauss–Bonnet–Chern integrand is a quadratic expression $\frac{1}{32\pi^2}(|R|^2-4|\mathrm{Ric}|^2+S^2)$ in the curvature tensor that is pointwise nonnegative under either curvature sign, giving $\chi\geq 0$ when $K\leq 0$ and $\chi>0$ when $K<0$ (Chern 1955). In dimensions $2n\geq 6$ this local route is blocked: Geroch (1976) and Klembeck (1976) exhibited metrics of positive sectional curvature whose Gauss–Bonnet–Chern integrand takes negative values somewhere, so only the integral, not the integrand, can carry the conjectured sign.

The principal global strategy, suggested by I. M. Singer and formulated as a conjecture of Dodziuk–Singer, works on the universal cover: by the Cartan–Hadamard theorem, a closed manifold of $K\leq 0$ has contractible universal cover $\tilde M$, and the $L^2$-Betti numbers $b^{(2)}_p(\tilde M)$ — the von Neumann dimensions of the spaces of $L^2$ harmonic $p$-forms on $\tilde M$, where the fundamental group $\Gamma=\pi_1(M)$ acts — satisfy Atiyah's covering index theorem $\chi(M)=\sum_p(-1)^p b^{(2)}_p(\tilde M)$. If all $b^{(2)}_p$ with $p\neq n$ vanished (the Singer conjecture for nonpositively curved manifolds, open in general), the sign $(-1)^n\chi(M)=b^{(2)}_n(\tilde M)\geq 0$ would follow. Confirmed cases all exploit extra structure: Gromov (1991) introduced Kähler hyperbolicity — the condition that the Kähler form (the symplectic form of a complex, hermitian metric) on the universal cover is $d$ of a bounded 1-form — and used the hard Lefschetz theorem (the isomorphisms $[\omega]^{k}:H^{n-k}\to H^{n+k}$ on Kähler cohomology) to prove $(-1)^n\chi>0$ for Kähler hyperbolic manifolds, which include all compact Kähler manifolds of $K<0$; Jost–Zuo (2000) and Cao–Xavier (2001) independently handled the nonpositive Kähler case via Kähler non-ellipticity/parabolicity; Chen–Yang (2017, 2021) extended this to Kähler manifolds with hyperbolic, CAT(0), or automatic fundamental group; Donnelly–Xavier (1984) proved the relevant $L^2$ vanishing for sufficiently pinched negative curvature, which yields the sign for compact quotients under pinching; Tan–Wang–Zhou (2019) and Huang–Tan (2021) treat symplectic manifolds satisfying a hard-Lefschetz-type condition (the $L^2$-hard Lefschetz property). A parallel combinatorial analogue, the Charney–Davis conjecture (1995), concerns the Euler characteristic of nonpositively curved piecewise Euclidean spaces and drives a separate program. The positive-curvature twin (Hopf's conjecture that $K>0$ implies $\chi>0$) is likewise open in dimensions $\geq 6$, with confirmed cases under torus symmetry (Kennard 2013; Kennard–Mouillé–Nienhaus 2025).

Problem Statement

Let $M$ be a closed Riemannian manifold of even dimension $2n$.

(i) If all sectional curvatures of $M$ satisfy $K\leq 0$, must $(-1)^n\chi(M)\geq 0$?

(ii) If all sectional curvatures satisfy $K<0$ strictly, must $(-1)^n\chi(M)>0$?

Both statements are classical for $n\leq 2$ (dimensions 2 and 4). The problem is to decide them in general for $n\geq 3$ (dimensions $\geq 6$), for arbitrary smooth Riemannian metrics, with no Kähler, complex, symplectic, hard-Lefschetz, curvature-pinching, or symmetry hypotheses added.

References

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