mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf. The Hopf conjecture...

Heinz Hopf in 1926 after determining the fundamental groups of three-dimensional spherical space forms as a generalization of the Poincaré conjecture to...

Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Hopf was born...

notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known...

length The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds Osserman conjecture: that every...

conjecture Kelvin's conjecture Kouchnirenko's conjecture Mertens conjecture Pólya conjecture, 1919 (1958) Ragsdale conjecture Schoenflies conjecture (disproved...

phrase non-existence as a conjecture. He also established the conjecture for perturbations of the Hopf fibration. The conjecture was disproven in 1974 by...

field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee...

pairs, answering a question posed in 1934 by Heinz Hopf and Erika Pannwitz. Andrew Vázsonyi conjectured bounds on the number of diameter pairs in higher...

of compact positively curved manifolds, leaving a lot of conjectures (e.g., the Hopf conjecture on whether there is a metric of positive sectional curvature...

This result is known as the differentiable sphere theorem. Heinz Hopf conjectured that a simply connected manifold with pinched sectional curvature is...

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly...

Witten–Reshetikhin–Turaev invariants (WRT-invariants). Let A {\displaystyle A} be a ribbon Hopf algebra over a field k {\displaystyle \Bbbk } (one can take, for example...

{\displaystyle H\neq 0} must be a standard sphere. Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature...

definition for an index of an umbilic point, and the global conjecture follows by the Poincaré–Hopf index theorem. No paper was submitted by Cohn-Vossen to...

scholars, endorses Raymond-Joseph Loenertz's refutation of Hopf's conjecture. He writes that Hopf misinterpreted Marino Sanudo's report of her death and its...

the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's theorem, and the Poincaré conjecture. Differential topology considers...

solvability of statistical mechanical models. He also generalized Hopf algebras to quasi-Hopf algebras and introduced the study of Drinfeld twists, which can...

discovered by Henry C. Wente (1986). It is a counterexample to the conjecture of Heinz Hopf that every closed, compact, constant-mean-curvature surface is...

giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of S3 as a subset of C2, the action is given by ( z...

Orsini's rule is not based on documentary evidence, but on Karl Hopf's conjectures, and A. Kiesewetter proposes that Maio di Monopoli (alias Matthew...

defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935). Whitehead's original homomorphism is defined geometrically, and gives...

This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables...

hyperbolic space, although a space form need not be simply connected. The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional...

theorem Saddle point Morse theory Lie derivative Hairy ball theorem Poincaré–Hopf theorem Stokes' theorem De Rham cohomology Sphere eversion Frobenius theorem...

it as yet unproven so is known as the sharp five exponentials conjecture. This conjecture implies both the sharp six exponentials theorem and the five...

known for Sweedler's Hopf algebra, Sweedler's notation, measuring coalgebras, and his proof, with Harry Prince Allen, of a conjecture of Nathan Jacobson...

nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The...

Gordon conjectured these were the only knots admitting lens space surgeries. This is now known as the Berge conjecture. The Berge conjecture states that...

obtained results similar to Fatou's. In 1916 he formulated the Bieberbach conjecture, stating a necessary condition for a holomorphic function to map the open...