Hopf sign conjecture

The Hopf sign conjecture asserts that nonpositive curvature controls the sign of the Euler characteristic: a closed connected Riemannian manifold $M^{2m}$ with sectional curvature $K\le0$ should satisfy $(-1)^m\chi(M)\ge0$. The expectation goes back to Hopf's lectures around 1931 on curvature and vector fields [Hopf1931CurvatureEuler], though the exact first written formulation is historically uncertain.

In dimension $2$ the statement follows at once from the Gauss–Bonnet theorem, and in dimension $4$ it follows from the Chern–Gauss–Bonnet formula, whose integrand is pointwise nonnegative under $K\le0$ there; this pointwise argument does not extend to dimension six and higher. Beyond low dimensions the conjecture is known for various locally symmetric and other high-symmetry classes, and the Euler characteristic of nonpositively curved manifolds has been studied more broadly [Davis1995EulerNonpositive]. Recent work examines the borderline case of nonpositively curved $4$-manifolds with vanishing Euler characteristic [ConnellRuanWang2026Euler].

The conjecture is open in every even dimension at least $6$; a resolution would require a mechanism relating sectional curvature to the Euler characteristic that avoids the sign analysis of the Gauss–Bonnet integrand.