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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 [[differential geometry of surfaces|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

: <math> k-(\operatorname{rank} G-\operatorname{rank} H)\leq 5. </math>

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.

== Aspherical manifolds ==
: ''Suppose ''M''2''k'' is a closed, [[aspherical manifold|aspherical]] manifold of even dimension. Then its Euler characteristic satisfies the inequality''

:: <math> (-1)^k\chi(M^{2k})\geq 0. </math>

This topological version of Hopf conjecture for [[Riemannian manifold]]s 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 <ref>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</ref>

== References ==
<references/>
* Thomas Püttmann and Catherine Searle, [http://www.ams.org/proc/2002-130-01/S0002-9939-01-06039-7/S0002-9939-01-06039-7.pdf ''The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank''], Proc AMS, 130:1 (2001), pp 163–166

[[Category:Differential geometry]]
[[Category:Topology]]
[[Category:Conjectures]]

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