Contrast resolution

In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to 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 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

$k-(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}G-\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}H)\le 5.$

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 S²×S² should admit no Riemannian metric with positive sectional curvature.

Aspherical manifolds

Suppose M²k is a closed, aspherical manifold of even dimension. Then its Euler characteristic satisfies the inequality

$(-1{)}^k\chi (M^{2k})\ge 0.$

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.

Metrics with no conjugate points

A Riemannian metric without conjugate points on n-dimensional torus is flat."

Proved by D. Burago and S. Ivanov

References

• Thomas Püttmann and Catherine Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proc AMS, 130:1 (2001), pp 163–166