Instability of the geodesic flow for the energy functional (Q627406)

The main result proved in the paper asserts that the geodesic flow of the unit tangent bundle of the unit sphere \(S^{n}(r)\), gives an unstable map \(T_{1}S^{n}(r)\rightarrow T_{1}T_{1}S^{n}(r)\) for \(n\geq 7\) and \(r>0\), where the metrics on the unit tangent bundles are the Sasaki ones. This gives a positive answer to a problem raised by \textit{E. Boeckx, J. C. González-Dávila} and \textit{L. Vanhecke} in [Commentat. Math. Univ. Carol. 43, No.~2, 201--213 (2002; Zbl 1090.53035)]. The author proves also that, for the unit sphere \((r=1)\), the result is invariant under a four-parameter deformation of the Sasaki metric on \(T_{1}T_{1}S^{n}(r)\).