Vol. 249, No. 2, 2011

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Instability of the geodesic flow for the energy functional

Domenico Perrone

Vol. 249 (2011), No. 2, 431–446

Let (Sn(r),g0) be the canonical sphere of radius r. Denote by Gs the Sasaki metric on the unit tangent bundle T1Sn(r) induced from g0 and by Gs the Sasaki metric on T1 T1Sn(r) induced from Gs. We resolve here, for n 7, a question raised by Boeckx, González–Dávila, and Vanhecke: namely, we prove that the geodesic flow

ξ : (T Sn(r), ^G ) → (T T Sn(r), ^^G )
1       s     1 1       s

is an unstable harmonic vector field for any r > 0 and n 7. In particular, in the case r = 1, ξ is an unstable harmonic map. We show that these results are invariant under a four-parameter deformation of the Sasaki metric Gs.

geodesic flow, canonical sphere, stability, energy functional, harmonic maps, natural Riemannian metrics
Mathematical Subject Classification 2000
Primary: 53C43, 53D25
Received: 9 February 2010
Accepted: 8 November 2010
Published: 1 February 2011
Domenico Perrone
Dipartimento di Matematica “E. De Giorgi”
Università del Salento
I-73100 Lecce