Hopf conjecture (Euler characteristic)

The Hopf conjecture asserts that every compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. It remains unresolved in general.

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Sectional curvature: A measure of Riemannian curvature assigned to each two-dimensional tangent plane at a point. The conjecture assumes this curvature is positive on every tangent plane.
Round sphere: A sphere with its standard metric of constant positive sectional curvature. Its Euler characteristic is two, matching the conjectured positivity.
Heinz Hopf: A Swiss mathematician whose work helped establish modern differential geometry and topology. The conjecture is named for his question connecting positive curvature with Euler characteristic.
Hopf conjecture (product curvature): The conjecture that the product of two equal-dimensional spheres admits no metric of positive sectional curvature. It is a separate, sharper obstruction question associated with Hopf, rather than the Euler-characteristic prediction.
Hopf conjecture in dimension four: The four-dimensional case of the Hopf Euler-characteristic conjecture. This prominent unresolved case asks whether positive sectional curvature forces positive Euler characteristic in dimension four.
Euler characteristic: A topological invariant equal to the alternating sum of the ranks of a space’s homology groups. Its sign is the quantity the conjecture predicts.
Complex projective space: A compact manifold of complex lines through the origin in a complex vector space. With the Fubini–Study metric, it has positive sectional curvature and positive Euler characteristic.
Wilhelm Klingenberg: A German differential geometer known for work on geodesics and global Riemannian geometry. His sphere theorem established strong topological consequences under pinched positive curvature.
Bonnet–Myers theorem: A theorem stating that complete manifolds with Ricci curvature bounded positively below are compact with bounded diameter. Positive Ricci curvature gives strong global conclusions, but is weaker than positive sectional curvature.
Chern–Gauss–Bonnet theorem: A theorem expressing the Euler characteristic of an even-dimensional Riemannian manifold as an integral of curvature expressions. Its integrand is not generally positive under positive sectional curvature in higher dimensions.