Instability of the geodesic flow for the energy functional

Abstract

Let .S n .r/; g 0 / be the canonical sphere of radius r. Denote by z G s the Sasaki metric on the unit tangent bundle T 1 S n .r/ induced from g 0 and by z G z s the Sasaki metric on T 1 T 1 S n .r/ induced from z G s . We resolve here, for n 7, a question raised by Boeckx, González-Dávila, and Vanhecke: namely, we prove that the geodesic flowis an unstable harmonic vector field for any r > 0 and n 7. In particular, in the case r D 1, is an unstable harmonic map. We show that these results are invariant under a four-parameter deformation of the Sasaki metric z G z s .