A criterion for local instability of a geodesic flow from the Jacobi equation in Fermi basis (Q1185420)

The authors consider the geodesic flow on Riemannian manifolds. They investigate the geodesic deviation equation in the Fermi basis. It appears that the averaged deviation equation written in the gradient form is defined by the Ricci scalar \(R\) in such a way that it is unstable if \(R<0\). The authors conclude that the negativity of \(R\) is a sufficient condition for the instability of geodesics. By means of the Maupertuis principle this result is applied to multidimensional homogeneous cosmological models.