Geodesics in the unit tangent bundle of a round sphere
type: Item
instance of: scholarly article
title: Geodesics in the unit tangent bundle of a round sphere
zbMATH Open document ID: 0675.53041
author: Herman Gluck
published in: L'Enseignement Mathématique. 2e Série
publication date: 1988-01-01T00:00:00Z
review text: 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 \(S^ n\).
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 4105784
zbMATH Keywords: geodesics
zbMATH Keywords: unit tangent bundle
zbMATH Keywords: round n-sphere
reviewed by: Andrew Bucki
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4105784