Hopf conjecture

Remarks

Both parts are open in dimension $6$ and higher. In dimension $2$ both parts follow from the Gauss–Bonnet theorem, since the integral of the Gaussian curvature equals $2\pi \chi(M)$. In dimension $4$ both parts follow from the Chern–Gauss–Bonnet theorem, because in that dimension the integrand is positive when the sectional curvature is either everywhere positive or everywhere negative. For the positive case in dimension $4$ there is also a topological argument: by Synge's theorem an orientable such $M$ is simply connected, so $b_1 = b_3 = 0$ and $\chi(M) = 2 + b_2 > 0$. In dimensions $2d \ge 6$ the Chern–Gauss–Bonnet integrand can be negative at a point even when the sectional curvature is positive, so this route fails. The positive case is known for manifolds whose curvature is pinched enough, since then the sphere theorem shows that $M$ is covered by a sphere. Weaker forms with non-strict inequalities are also attributed to Hopf: nonnegative curvature should give $\chi(M) \ge 0$, and nonpositive curvature should give $(-1)^d \chi(M) \ge 0$. A different question of Hopf, often also called the Hopf conjecture, asks whether $S^2 \times S^2$ carries a metric of positive sectional curvature; this is also open.