Hopf Conjecture
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High Quality Content by WIKIPEDIA articles! In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.For surfaces, this follows from the Gauss Bonnet theorem. For four-dimensional manifolds, this follows from the finiteness of the fundamental group and the Poincaré duality. In particular, the four-dimensional manifold S2×S2 should admit no Riemannian metric with positive sectional curvature.This topological version of Hopf conjecture for Riemannian manifolds is due to William Thurston. Ruth Charney and…mehr

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High Quality Content by WIKIPEDIA articles! In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.For surfaces, this follows from the Gauss Bonnet theorem. For four-dimensional manifolds, this follows from the finiteness of the fundamental group and the Poincaré duality. In particular, the four-dimensional manifold S2×S2 should admit no Riemannian metric with positive sectional curvature.This topological version of Hopf conjecture for Riemannian manifolds is due to William Thurston. Ruth Charney and Mike Davis conjectured that the same inequality holds for a nonpositively curved piecewise Euclidean (PE) manifold.