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 (T_1S^3, G_s), where G_s is the Sasaki metric. In this paper, by using a different method, we get (Theorem 1.1) an analogue of Han and Yim’s theorem by replacing the unit sphere S^3 by a Riemannian three-manifold of constant sectional curvature k ≠0 and the Sasaki metric G_s by an arbitrary Riemannian g-natural metric G that is a deformation depending on three real parameters of the Sasaki metric G_s ; such a deformation preserves the property (of the Sasaki metric) that horizontal and vertical lifts are orthogonal. We do not assume that M is compact. So, in particular, M may be an open (connected) subset of the sphere S^3. In the case of the hyperbolic space H^n(−k) for k > 0, it is an open question whether some unit vector field exists (of course, non-Killing) that defines a harmonic map from H^n(−k) to (T_1H^n(−k), G_s). An immediate consequence of our result is that there does not exist a unit vector field on the hyperbolic three-space that defines a harmonic map. Such a result is invariant under a three parameter deformation of the Sasaki metric on T_1H^3(−k). We also extend Han and Yim’s theorem for Riemannian (2n + 1)-manifolds (M, g) of constant sectional curvature k > 0 with π_1(M) ≠0.
Unit vector fields on real space forms which are harmonic maps
PERRONE, Domenico
2009-01-01
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 S^3 that define harmonic maps from S^3 to (T_1S^3, G_s), where G_s is the Sasaki metric. In this paper, by using a different method, we get (Theorem 1.1) an analogue of Han and Yim’s theorem by replacing the unit sphere S^3 by a Riemannian three-manifold of constant sectional curvature k ≠0 and the Sasaki metric G_s by an arbitrary Riemannian g-natural metric G that is a deformation depending on three real parameters of the Sasaki metric G_s ; such a deformation preserves the property (of the Sasaki metric) that horizontal and vertical lifts are orthogonal. We do not assume that M is compact. So, in particular, M may be an open (connected) subset of the sphere S^3. In the case of the hyperbolic space H^n(−k) for k > 0, it is an open question whether some unit vector field exists (of course, non-Killing) that defines a harmonic map from H^n(−k) to (T_1H^n(−k), G_s). An immediate consequence of our result is that there does not exist a unit vector field on the hyperbolic three-space that defines a harmonic map. Such a result is invariant under a three parameter deformation of the Sasaki metric on T_1H^3(−k). We also extend Han and Yim’s theorem for Riemannian (2n + 1)-manifolds (M, g) of constant sectional curvature k > 0 with π_1(M) ≠0.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
