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.
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 4k+2 or 4k+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".
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.
Proved by D. Burago and S. Ivanov