Hopf conjecture

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 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

$k-(\operatorname{rank} G-\operatorname{rank} H)\leq 5.$

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

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

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 ^[1]

References

1. D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, GEOMETRIC AND FUNCTIONAL ANALYSIS Volume 4, Number 3 (1994), 259-269, DOI: 10.1007/BF01896241

• 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