Acta Math. Hungar., 113 (1–2) (2006), 51–62. CLAIRAUT RELATION FOR GEODESICS OF HOPF TUBES

Abstract. In this note we use the Hopf map pi: S3 → S2 to construct an in-teresting family of Riemannian metrics hf 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 pi−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 Clairaut relation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible. 1