Clairaut relation for geodesics of Hopf tubes (Q879188)

The theorem of L. C. Clairaut is known: If \(\theta\) is the angle between the tangent to the curve and a circle of latitude, and if \(r\) is the radius of this circles, then \(r\cos \theta =\) const. along the curve. By using the hopf maps \(\pi:S^3\rightarrow S^2\), the authors have constructed a Riemannian metrics \(h^f\) on the 3-sphere. Then the authors considered the Hopf tube over an immersed curve \(\nu\) in \(S^2\) is the complete lift \(\pi^{-1}(\nu)\) of \(\nu\), and it is proved that any geodesic of this Hopf tube satisfies the Clairaut relation. Also it is proved that if the sphere \(S^3\) is equipped with a familly \(h^f\) of Lorentzian metrics, then it is obtained a new Clairaut relation for timelike geodesics of the Lorentzian Hopf tube.