Geodesics in the unit tangent bundle of a round sphere
The author determines the geodesics in the unit tangent bundle \(US^ n\) of the round n-sphere \(S^ n\). Let p(t) be a spherical helix lying in a great 3-sphere \(S^ 3\) inside \(S^ n\) and consider the Frenet frame \(\{T(t),N(t),B(t)\}\) along p(t) and the Frenet equations: \(T'=\kappa N\), \(N=-\kappa T-\tau B\), \(B=\tau N\), where \(\kappa\) denotes curvature and \(\tau\) torsion, and let v(t) be a unit vector field along p(t) which appears constant in Frenet coordinates. Then, the author proves that (p(t),v(t)) is a geodesic in \(US^ n\) if and only if \(\sqrt{\kappa^ 2+\tau^ 2}=\| v'(t)\| /\| p'(t)\|\) where \(p'(t)\) is a tangent vector to p(t) and \(v'(t)\) is the covariant derivative of v(t) along p(t) and the norms of these vectors are measured in the given Riemannian metric of \(S^ n\).
scientific article; zbMATH DE number 4105784