Clairaut relation for geodesics of Hopf tubes.

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. [ABSTRACT FROM AUTHOR]