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...

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...

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...

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

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...

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...

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...

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....

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...

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...

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...

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...

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

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...

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...

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

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 ≃...

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...

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

, 1 ) = 1 {\displaystyle Q(1,1)=1} : a (1,1) self-links, as in the Hopf fibration. This form thus has Arf invariant 0 (most of its elements have norm...

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...

horocycle flow on a hyperbolic manifold. This can be seen to be a kind of Hopf fibration. Such flows commonly occur in classical mechanics, which is the study...

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...

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

not all preimages of magnetisation have a nonzero linking. Skyrmion Hopf fibration Faddeev L, Niemi AJ (1997). "Stable knot-like structures in classical...