The name Hopf conjecture is used for two different conjectures of Heinz Hopf relating the topology of a Riemannian manifold to its sectional curvature.
The sign form asserts that every compact manifold of even dimension equipped with a Riemannian metric having everywhere positive sectional curvature has positive Euler characteristic. A corresponding nonnegative form predicts a nonnegative Euler characteristic when the sectional curvature is everywhere nonnegative.
The product form asserts that no direct product of two closed manifolds of positive dimension
admits a Riemannian metric having everywhere
positive sectional curvature. Its best-known
special case concerns 
.
Here each 
is an ordinary 2-dimensional sphere, and 
consists of ordered
pairs 
of points, one from each sphere.
It is a closed manifold of dimension
4. Its standard product Riemannian metric has
nonnegative sectional curvature: two-planes
tangent to a sphere factor have positive sectional
curvature, but mixed two-planes have zero sectional
curvature. Cheeger's deformation method
gives other Riemannian metrics with nonnegative
sectional curvature on 
, studied further by Müter (1987), but two-planes
of zero sectional curvature remain. Hsiang
and Kleiner (1989) proved that a closed manifold
of dimension 4 with positive sectional
curvature and a nontrivial Killing vector
must be homeomorphic to the 4-sphere
or the complex projective plane. Consequently,
a Riemannian metric on 
with positive sectional
curvature must have no continuous symmetry.
Brendle and Hung (2026) announced a construction of a Riemannian metric with positive sectional curvature
on 
.
Their proposed construction starts from a Cheeger-Müter
metric and applies a third-order perturbation, with some supporting symbolic
calculations performed in the Wolfram Language. If the proof
is verified, it disproves the product form of the Hopf conjecture, though it does
not affect the sign form, since the Euler characteristic
of 
is 4.
with Positive Sectional Curvature." 19 Aug
2026.