Resumen de Harmonic maps defined by the geodesic flow

Mohamed T. K. Abbassi

• Let (M,g) be a Riemannian manifold. We equip the unit tangent sphere bundle T1 M of (M,g) and its unit tangent sphere bundle Tr T1M of radius r>0 with arbitrary Riemannian g-natural metrics. When (M,g) is two-point homogeneous and both T1 M and T1T1M are equipped with the Sasaki metrics, the geodesic flow vector field is harmonic and determines a harmonic map [E. Boeckx and L. Vanhecke, Harmonic and minimal vector fields on tangent and unit tangent bundles, Diff. Geom. Appl., 13 (2000), 77-93]. We prove that if arbitrary Riemannian g-natural metrics are considered, then the geodesic flow is still a harmonic vector field, and it also defines a harmonic map under some conditions on the g-natural metrics. This permits to exhibit large families of harmonic maps defined in a compact Riemannian manifold and having a target space with a highly nontrivial geometry. In particular, explicit examples are provided on the unit tangent sphere bundle of the sphere S n and the flat torus Tn. Moreover, the geodesic flow being a Killing vector field is characterized in terms of harmonicity of the corresponding map and of properties of the base manifold.