## The Hopf Fibration and its Applications

### BibTeX

@MISC{Porter_thehopf,

author = {Mason A. Porter},

title = {The Hopf Fibration and its Applications},

year = {}

}

### OpenURL

### Abstract

this paper, I will discuss some of these applications as motivation for mathematicians to learn about the physical significance of the Hopf fibration in greater detail than that presented here. I will focus my attention primarily on rigid body rotation and the Hopf fibration h : S

### Citations

589 | Introduction to Mechanics and Symmetry
- Marsden, Ratiu
- 1999
(Show Context)
Citation Context ...ifold structure in the sense that there exists a transformation �� : G��G -# G, where �� is C # ("smooth") and maps (g, h) ## gh. Examples of Lie groups include SO(n), SL(n, R), =-=and SU(n) (please see [16]-=- or [22] for more detail). To each Lie group G is associated a Lie algebra g. . A Lie algebra g is a vector space along with a bilinear anti-symmetric bracket [#, #] that satisfies Jacobi's identity: ... |

229 |
Polycyclic Groups
- Segal
- 1983
(Show Context)
Citation Context ...ch as Einstein's theory of gravity (see [2]). A second lies in the study of singularities in algebraic geometry (see [3] and [11]), and a third application is in the theory of integrable systems (see =-=[19]-=-). 13 Coadjoint Orbits Coadjoint orbits are group orbits for the action of G on g # . Background: Group Actions . Let G be a group and Y a set. The group G then acts on Y if there exists a continuous ... |

213 | Abstract algebra - Dummit, Foote - 1999 |

147 | Theorie der Transformationsgruppen - Lie, Engel |

83 | An Introduction to Algebraic Topology - Rotman - 1993 |

67 |
The Topology of Fiber Bundles
- Steenrod
- 1951
(Show Context)
Citation Context ...ctor space over the real quaternions H. Setting n = 2, we obtain the map h 2 : S 15 -# OP 2 # = S 8 . This map has typical fiber S 7 . (Interested readers should examine page 54 of [7] or page 108 of =-=[21]-=- for more information.) Let us now speculate further about higher dimensional Hopf fibrations. In our previous constructions, the reader may have noticed the use of division algebras of dimensions 1 (... |

50 |
Clifford Algebras and the Classical Groups, Cambridge
- Porteous
- 1995
(Show Context)
Citation Context ...tic conditions. Using the Cayley numbers, we find yet another Hopf fibration series, hn : S 6n+3 -# OP n . Background: Cayley Numbers . The Cayley numbers O, whose properties and definition appear in =-=[18]-=-, form a nonassociative real division algebra with additive group R 8 . This division algebra is a two dimensional vector space over the real quaternions H. Setting n = 2, we obtain the map h 2 : S 15... |

43 |
Classical Mechanics (Second Edition
- GOLDSTEIN
- 1980
(Show Context)
Citation Context ...) {F, G} = -{G,F} (anti-symmetry). (iii) The Jacobi identity (defined earlier) is satisfied. (iv) {FG,H} = F{G,H} + {F, H}G (Liebniz identity). 8.3 General Poisson Brackets In classical texts such as =-=[6]-=-, the map from material representation to body coordinates is given by the association: (#, #, #, P # , P # , P# ) ## (# 1 , # 2 , # 3 ), defined by: # 1 = 1 sin # [(P # - P# cos #) sin # + P # sin # ... |

35 |
The structure of the space of solutions of Einstein’s equations I: One Killing
- Fischer, Marsden, et al.
- 1980
(Show Context)
Citation Context ...ects of mechanics---we mention three of them in passing. The first is in the study of the theory of the space of all solutions in a relativistic field theory such as Einstein's theory of gravity (see =-=[2]-=-). A second lies in the study of singularities in algebraic geometry (see [3] and [11]), and a third application is in the theory of integrable systems (see [19]). 13 Coadjoint Orbits Coadjoint orbits... |

33 |
Foundations of di¤erentiable manifolds and Lie groups
- Warner
- 1971
(Show Context)
Citation Context ...ructure in the sense that there exists a transformation �� : G��G -# G, where �� is C # ("smooth") and maps (g, h) ## gh. Examples of Lie groups include SO(n), SL(n, R), and SU(n=-=) (please see [16] or [22]-=- for more detail). To each Lie group G is associated a Lie algebra g. . A Lie algebra g is a vector space along with a bilinear anti-symmetric bracket [#, #] that satisfies Jacobi's identity: [[#, #],... |

27 | Uber die Abbildungen der dreidimensionalen Sphare auf die Kugel - Hopf - 1931 |

14 | Angular momentum, convex polyhedra and algebraic geometry
- Atiyah
- 1983
(Show Context)
Citation Context ...study of the theory of the space of all solutions in a relativistic field theory such as Einstein's theory of gravity (see [2]). A second lies in the study of singularities in algebraic geometry (see =-=[3]-=- and [11]), and a third application is in the theory of integrable systems (see [19]). 13 Coadjoint Orbits Coadjoint orbits are group orbits for the action of G on g # . Background: Group Actions . Le... |

10 |
An Introduction to Homotopy Theory
- Hilton
- 1953
(Show Context)
Citation Context ... two dimensional vector space over the real quaternions H. Setting n = 2, we obtain the map h 2 : S 15 -# OP 2 # = S 8 . This map has typical fiber S 7 . (Interested readers should examine page 54 of =-=[7]-=- or page 108 of [21] for more information.) Let us now speculate further about higher dimensional Hopf fibrations. In our previous constructions, the reader may have noticed the use of division algebr... |

6 |
Topology, geometry, and gauge fields. Foundations
- Naber
- 1997
(Show Context)
Citation Context ...Hopf fibration g 1 : S 7 -# HP 1 # = S 4 . The "natural connection" of this Hopf bundle represents a BPST (Belavin, Polyakov, Shwartz, and Tyupkin) instanton solution to the Yang-Mills equat=-=ions (see [17] for more -=-details). The BPST instantons were initially termed "pseudoparticles." They were not viewed as the coordinate expression for a globally defined connection on a bundle. Now, however, BPST ins... |

5 | Elementary Classical Analysis. Second edition, W.H.Freeman and - Marsden, Hoffman - 1993 |

2 |
Raibert [1990] Biped gymnastics
- Hodgins, M
(Show Context)
Citation Context ...portant in the field of robotics. One can see this in light of the following example: a team at the MIT artificial intelligence laboratory is attempting the construction of a somersaulting robot (see =-=[9]-=-). More specifically, the project entails the fabrication of a robot that will get a running start before launching itself into the air, completing a forward revolution, and ultimately landing firmly ... |

1 |
1982] Reduction of the Semi-simple 1:1
- Cushman, Rod
(Show Context)
Citation Context ...2 , p 2 )} -# {(w 1 , w 2 , w 3 )}. When restricted to level sets, it maps S 3 to S 2 . Oscillatory systems in 1:1 resonance are, in fact, related to rigid body rotation. Indeed, Cushman and Rod (see =-=[4]-=-) have shown how to analyze the 1:1 resonance using rigid body dynamics. 5 Rigid Body Dynamics Before we discuss rigid body rotation in detail, we must first consider some terminology and definitions ... |

1 | Mechanics with Computer Graphics: Linear Hamiltonian Systems - Topology |