Unit vector fields on real space forms which are harmonic maps

Domenico Perrone

Vol. 239 (2009), No. 1, 89–104

DOI: 10.2140/pjm.2009.239.89

Abstract

In 1998, Han and Yim proved that the Hopf vector fields, namely, the unit Killing vector fields, are the unique unit vector fields on the unit sphere S³ that define harmonic maps from S³ to (T¹S³,G_s), where G_s is the Sasaki metric. In this paper, by using a different method, we get an analogue of Han and Yim’s theorem for a Riemannian three-manifold with constant sectional curvature k≠0. An immediate consequence is that there does not exist a unit vector field on the hyperbolic three-space that defines a harmonic map. We also extend this result for Riemannian (2n + 1)-manifolds (M,g) of constant sectional curvature k > 0 with π₁(M)≠0.

Keywords

harmonic maps, unit Killing vector fields, real space forms, Riemannian g-natural metrics.

Mathematical Subject Classification 2000

Primary: 58E20, 53C43

Milestones

Received: 11 June 2008

Accepted: 2 September 2008

Published: 1 January 2009