1301.6352 (Ricardo Gallego Torromé)

On k-jet field approximations of geodesic deviation equations    [PDF]

Ricardo Gallego Torromé

Let M be a smooth manifold and S a spray defined on the convex cone C of the tangent bundle TM. It is proved that the only non-trivial k-jet approximation of the exact geodesic deviation equation of S, linear on the deviation functions and invariant under arbitrary local coordinate transformations corresponds to the Jacobi equation. However, if linearity in the deviation functions is not required, there are differential equations whose solutions admit k-jet approximations and are invariant under arbitrary coordinate transformations. As an example of higher order geodesic deviation equations we study the first and second order jet geodesic deviation equations for a Finsler spray.

View original: http://arxiv.org/abs/1301.6352