UNIT VECTOR FIELDS ON REAL SPACE FORMS WHICH ARE HARMONIC MAPS

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 S-3 that define harmonic maps from S-3 to ((TS3)-S-1, (G) over tilde (s)), where (G) over tilde (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 not equal 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 pi(1)(M) not equal 0.