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- Title
Clairaut relation for geodesics of Hopf tubes.
- Authors
Cabrerizo, J. L.; Fernández, M.
- Abstract
In this note we use the Hopf map π : S³ → S² to construct an interesting family of Riemannian metrics h f on the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π-1 (γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R³. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family hf of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.
- Subjects
MATHEMATICS; GEODESICS; DIFFERENTIAL geometry; RIEMANNIAN submersions; MATHEMATICAL functions; RIEMANNIAN geometry
- Publication
Acta Mathematica Hungarica, 2006, Vol 113, Issue 1/2, p51
- ISSN
0236-5294
- Publication type
Article
- DOI
10.1007/s10474-006-0089-6