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In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to either Eberhard Hopf or Heinz Hopf. == Positively curved Riemannian manifolds == : ''A compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic'' 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. The conjecture has been proved for manifolds of dimension 4''k''+2 or 4''k''+4 admitting an isometric torus action of a ''k''-dimensional torus and for manifolds ''M'' admitting an isometric action of a compact Lie group ''G'' with principal isotropy subgroup ''H'' and cohomogeneity ''k'' such that : In a related conjecture, "positive" is replaced with "nonnegative". == Riemannian symmetric spaces == : ''A compact symmetric space of rank greater than one cannot carry a Riemannian metric of positive sectional curvature.'' In particular, the four-dimensional manifold ''S''2×''S''2 should admit no Riemannian metric with positive sectional curvature. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hopf conjecture」の詳細全文を読む スポンサード リンク
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