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VOL. 7 | 2006 On Special Types of Minimal and Totally Geodesic Unit Vector Fields
Alexander Yampolsky

Editor(s) Ivaïlo M. Mladenov, Manuel de León

Abstract

We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasakian metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in $T_{1}S^{n}$.

Information

Published: 1 January 2006
First available in Project Euclid: 14 July 2015

zbMATH: 1095.53026
MathSciNet: MR2228380

Digital Object Identifier: 10.7546/giq-7-2006-292-306