In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) C 2 ∖ { 0 } {\displaystyle...

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...

mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important...

compact complex manifolds in general, as shown by the example of the Hopf surface, which is diffeomorphic to S1 × S3 and hence has b1 = 1. The "Kähler...

compact complex manifolds in general, as shown by the example of the Hopf surface, which is diffeomorphic to S1 × S3 and hence has b1 = 1. The "Kähler...

arbitrarily small neighbourhood of 0. Two-dimensional Hopf manifolds are called Hopf surfaces. In a typical situation, Γ {\displaystyle \Gamma } is generated...

irregularity q is 1, and h1,0 = 0. All plurigenera are 0. Hodge diamond: Hopf surfaces are quotients of C2−(0,0) by a discrete group G acting freely, and have...

points Bordiga surfaces, the White surfaces determined by families of quartic curves Vanishing second Betti number: Hopf surfaces Inoue surfaces; several other...

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...

hyperkähler manifold is also hypercomplex. The converse is not true. The Hopf surface ( H ∖ 0 ) / Z {\displaystyle {\bigg (}{\mathbb {H} }\backslash 0{\bigg...

Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be...

Teleman that any surface of class VII with b 2 = 0 {\textstyle b_{2}=0} is a Hopf surface or an Inoue-type solvmanifold. These surfaces have no meromorphic...

embedded surface of constant Gaussian curvature must be a sphere (Liebmann 1899). Heinz Hopf showed in 1950 that a closed embedded surface with constant...

topological construction based on Clifford parallel pointed out by Heinz Hopf (1931) The lines on 1 in elliptic space are described by versors with a fixed...

P1. These surfaces are never algebraic or Kähler. The minimal ones with b2 = 0 have been classified by Bogomolov, and are either Hopf surfaces or Inoue...

non-proper and non-separated algebraic spaces whose analytic space is the Hopf surface). It is also possible for different algebraic spaces to correspond to...

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

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...

In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks...

surfaces have small analytic deformations that are the blowups of primary Hopf surfaces at a finite number of points. In particular they have an infinite cyclic...

embedded surface in R 3 {\displaystyle \mathbb {R} ^{3}} with constant mean curvature H ≠ 0 {\displaystyle H\neq 0} must be a sphere, and H. Hopf proved...

there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it...

Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a German mathematician...

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

the Kähler form. It can fail for non-Kähler manifolds: for example, Hopf surfaces have vanishing second cohomology groups, so there is no analogue of...

presented at a congress in 1891. Hopf fibration Toric section Vesica piscis Coxeter 1969. Hirsch 2002. Dorst 2019, §6. Hopf Fibration and Stereographic Projection...

This latter pentagonal symmetry of the 600-cell is captured by the set of Hopf coordinates (𝜉i, 𝜂, 𝜉j) given as: ({<10}⁠𝜋/5⁠, {≤5}⁠𝜋/10⁠, {<10}⁠𝜋/5⁠)...

structure of a Z2-graded Hopf algebra. Likewise the representations of this Hopf algebra turn out to be Z2-graded comodules. This Hopf algebra gives the global...

conjecture of Heinz Hopf that every closed, compact, constant-mean-curvature surface is a sphere (though this is true if the surface is embedded). There...

is important in the study of S3 as a fiber bundle over S2 (the Hopf bundle). The surface described above, given the relative topology from R3, is homeomorphic...