In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)...

: 198  The Hopf fibration S 1 → S 3 → S 2 {\displaystyle S^{1}\to S^{3}\to S^{2}} is a non-trivial fiber bundle and, specifically, a Serre fibration. A mapping...

a Hopf link. This was Hopf's motivation for studying the Hopf link: because each two fibers are linked, the Hopf fibration is a nontrivial fibration. This...

time. At this time Hopf discovered the Hopf invariant of maps S 3 → S 2 {\displaystyle S^{3}\to S^{2}} and proved that the Hopf fibration has invariant 1...

The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial...

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

Example: the Hopf fibration. Let B equal S 2 {\displaystyle S^{2}} and E equal S 3 . {\displaystyle S^{3}.} Let p be the Hopf fibration, which has fiber...

All the fiber bundles are aspects of the same discrete Hopf fibration, because the fibration is the various expressions of the same distinct left-right...

d\xi _{1}\wedge d\xi _{2}.} To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above z 1 = e i ( ξ 1 +...

four-group is the smallest noncyclic group,... Shokurov, A.V. (2002). "Hopf fibration". In Michiel Hazewinkel (ed.). Encyclopedia of Mathematics. Helsinki:...

{\displaystyle d_{k}:E_{p,q}^{k}\to E_{p-k,q-1+k}^{k}} . Recall that the Hopf fibration is given by S 1 ↪ S 3 → S 2 {\displaystyle S^{1}\hookrightarrow S^{3}\to...

discrete Hopf fibration of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of...

skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell). The Petrie...

over the 2-sphere, known as Hopf fibration in the case r = i  ; other vectors give isomorphic, but not identical fibrations. Lyons (2003) gives an elementary...

space C 2 {\displaystyle \mathbb {C} ^{2}} to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere....

be regarded as a family of circles parametrized by CPn: this is the Hopf fibration. Complex projective space carries a (Kähler) metric, called the Fubini–Study...

singularities comes from considering the flow along the lines of a Hopf fibration. Let r {\textstyle r} be a constant radius of the inner coil. One set...

other naturally due to the Hopf fibration.[failed verification] The collective of edges forms another discrete Hopf fibration of 12 rings with 10 vertices...

Paul Dirac back in 1931 and Hopf fibration, a fibration of a 3-sphere proposed io the same year by mathematician Heinz Hopf. Mathematician Jim Simons discussing...

then the quotient map is a fiber bundle. One example of this is the Hopf fibration, S 3 → S 2 {\displaystyle S^{3}\to S^{2}} , which is a fiber bundle...

the circular fibers of a Hopf fibration SU(2) → S2. As such, the Berger metrics can also be constructed via the Hopf fibration, by scaling the tangent...

to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres. In algebraic geometry, an invertible sheaf (i...

Grids on SO(3) Using the Hopf Fibration". The International Journal of Robotics Research. 29 (7). Section 8 – Derivation of Hopf parametrisation. doi:10...

diffeomorphic manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration. Explicit expressions for coordinates...

trivial. Recall there is a spectral sequence for any Serre fibration, such as the fibration K ( π n + 1 ( X ) , n + 1 ) ≃ F n + 1 → S n + 1 n → S n n ≃...

known as the Bloch sphere or, equivalently, the Riemann sphere. See Hopf fibration for details of the projectivization construction in this case. The Cartesian...

CG]. Zamboj, Michal (8 January 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational...

"Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings". International Journal of Theoretical Physics. 16 (9): 561–565. Bibcode:1977IJTP...

action of the modular group. They also provide a canonical example of Hopf fibration, where the geodesic flow induced by the linear fractional transformation...

conjecture. He also established the conjecture for perturbations of the Hopf fibration. The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited...