Hopf conjecture

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In mathematics there two Hopf conjectures. They belong to the field of differential geometry, and are named for Heinz Hopf.

One of the Hopf conjectures states, that a compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. This is proved for dimension 2 (follows from the Gauss-Bonnet theorem) and 4 (where it follows from Poincaré duality). This conjecture is also proved in higher dimension n > 4, if you admit an isometric d-dimensional torus action, where

$d > \frac{n}{8}-\frac{1}{2}.$

The other Hopf conjecture states that

$S^2\times S^2$

admits no metric with positive sectional curvature.

A related, slightly more general question also posed by Hopf (to the latter conjecture) is the following: "Do compact even dimensional manifolds which admit at least one metric of positive sectional curvature necessarily have positive Euler characteristic?" Certainly if $S^2 \times S^2$ does not admit such a metric this provides a counterexample to the converse, because $S^2 \times S^2$ certainly has positive Euler characteristic. This question was posed, according to Berger, sometime in the early 1930s.