Hopf conjecture (on Euler characteristic and curvature)

E679323

mathematical conjecture open problem in differential geometry

The Hopf conjecture on Euler characteristic and curvature is an open problem in differential geometry proposing a deep link between the sign of a manifold’s Euler characteristic and the sign of its sectional curvature, especially for even-dimensional manifolds with positive or negative curvature.

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Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ open problem in differential geometry ⓘ
appliesTo	compact Riemannian manifold ⓘ even-dimensional compact manifold ⓘ
clarification	distinct from Hopf fibration conjectures ⓘ distinct from Hopf invariant one problem ⓘ
concerns	sign of Euler characteristic ⓘ sign of sectional curvature ⓘ
curvatureCondition	everywhere negative sectional curvature ⓘ everywhere positive sectional curvature ⓘ
dimensionCondition	even dimension ⓘ
field	Riemannian geometry NERFINISHED ⓘ differential geometry ⓘ global differential geometry ⓘ
hasAbbreviation	Hopf conjecture on Euler characteristic and curvature NERFINISHED ⓘ
hasVariant	Hopf conjecture for manifolds with negative sectional curvature NERFINISHED ⓘ Hopf conjecture for nonpositive curvature NERFINISHED ⓘ
implies	topological restrictions from curvature sign ⓘ
influenced	research on manifolds of positive curvature ⓘ research on pinched curvature ⓘ study of topological obstructions to curvature conditions ⓘ
motivation	understanding how curvature controls topology ⓘ
namedAfter	Heinz Hopf NERFINISHED ⓘ
namedEntityType	mathematical statement ⓘ
predicts	alternating sign of Euler characteristic for even-dimensional manifolds with negative sectional curvature ⓘ positive Euler characteristic for even-dimensional manifolds with positive sectional curvature ⓘ
proposedBy	Heinz Hopf NERFINISHED ⓘ
relatedConjecture	Hopf conjecture on product of spheres NERFINISHED ⓘ
relatedTo	Betti numbers NERFINISHED ⓘ Chern–Gauss–Bonnet theorem NERFINISHED ⓘ Euler characteristic NERFINISHED ⓘ Gauss–Bonnet theorem NERFINISHED ⓘ Hadamard–Cartan theorem NERFINISHED ⓘ Poincaré duality NERFINISHED ⓘ Riemannian manifold ⓘ even-dimensional manifold ⓘ homology sphere ⓘ negative sectional curvature ⓘ negatively curved manifold ⓘ positive curvature manifold ⓘ positive sectional curvature ⓘ sectional curvature ⓘ sphere theorem ⓘ
specialCaseOf	relationships between topology and curvature ⓘ
status	open ⓘ
studiedIn	global Riemannian geometry literature ⓘ
timePeriod	20th century ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Heinz Hopf → notableWork → Hopf conjecture (on Euler characteristic and curvature) ⓘ