Generating uniform incremental grids on so3 using the hopf fibration. Survey of hopf fibrations and rotation conventions in mathematics. An elementary introduction to the hopf fibration niles johnson. An elementary introduction to the hopf fibration by david w. The hopffibration and hidden variables in quantum and classical. Hyperbolic hopf fibrations the hopf fibration of s3 is amazing and beautiful. Differential and twistor geometry of the quantum hopf fibration simon brain and giovanni landi abstract.
We present in this article an introduction to the hopf fibration that requires only linear algebra and analytic geometry. Generalising the penrose twistor fibration mathoverflow. It describes the 3sphere by using circles inside of an ordinary sphere, the 2sphere. The twosphere is formed by all points which are a constant distance from a center point. The s2 base space of a suitably oriented s3 hopf fibration is nothing but the bloch sphere, while the circular fibres represent the qubit overall phase degree of freedom. The hopf fibration and its applications semantic scholar. The above example is a special case, for n1, since cp 1 is homeomorphic to s 2.
An elementary introduction to the hopf fibration david w. The hopf fibration describes rp3 in terms of a circle, s1, and an ordi nary 2sphere, s2. An interactive tool to explore rational elliptic curves in weierstrass form. The resulting foursphere s4 q is a quantum analogue.
The hopf brations also have the weaker symmetry property of being berwise homogeneous. The twosphere is the one youre likely to be familiar witha beach ball is a good example. The hopf fibration is a kind of projection from the threesphere to the twosphere. The topic names represent the content of the handouts. No installation, realtime collaboration, version control, hundreds of latex templates, and more. Lyons department of mathematical sciences lebanon valley college 101 n. Hopf fibrations and hurwitzradon numbers valentin ovsienko. Geometric interpretation using the complex projective lineedit. The matrix exponential pdf geometry of the special unitary group pdf the spectral theorem pdf symmetric forms pdf stereographic projection of the hopf fibration matlab program pdf courtesy of huan yao. Noteworthy fiber products with the quaternionic hopf fibration, notably exotic 7spheres, are discussed in llohann d. The hopf fibration s 1 s 3 s 2 was historically one of the earliest nontrivial examples of a fibration. Applied to the unit quaternion, the hopffibration is a projection between the. A brief introduction to the hopf fibration for introductory topology students.
See the history of this page for a list of all contributions to it. An example is a fibration of 3space by pairwise skew lines, the result. Mathematical gems and curiosities sergei tabachnikov, editor. S1equivariant dirac operators on the hopf fibration 3 we can read from the expressions above. Visualization in both 2d affine and 3d projective space.
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