Hopf conjecture refers to several conjectural statements in differential geometry and topology attributed to Heinz Hopf, concerning positively and negatively curved Riemannian manifolds. Hopf's questions first appeared in print from talks he gave in 1931, with a related statement allowing zero curvature following from a 1953 talk. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe modern formulation states two things: a compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic, and a compact, even-dimensional Riemannian manifold with negative sectional curvature has Euler characteristic whose sign matches negative one raised to half the manifold's dimension. 1 Proposed YearHeinz Hopf raised the underlying questions in 1931; the modern two-part formulation captured in the statement key is a later restatement of his original questions, per the same Wikipedia section. Progress Toward ResolutionThe conjecture is proved for surfaces, where it follows from the Gauss-Bonnet theorem, and a separate argument covers four-dimensional manifolds. For manifolds of dimension six or higher the conjecture remains open. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Wolfram MathWorld: Hopf Conjecture
Posed By
Source Wolfram MathWorld: Hopf Conjecture
Sources
1. Hopf Conjecture (Wikipedia)
Wikimedia FoundationLead section
Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf
Positively or negatively curved Riemannian manifolds section, second sentence
It goes back to questions by Heinz Hopf from 1931.
Positively or negatively curved Riemannian manifolds section, third sentence
For surfaces, these statements follow from the Gauss-Bonnet theorem.
View the Source Wolfram MathWorld: Hopf Conjecture
Wolfram MathWorldIn Branch: Differential Geometry, Subject classifications
Calculus and Analysis Differential Geometry
Posed By: Heinz Hopf, Lead
The name "Hopf conjecture" is used for two different conjectures of Heinz Hopf relating the topology of a Riemannian manifold to its sectional curvature .
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