# 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

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

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

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