On $k$-jet field approximations to geodesic deviation equations

TL;DR

This paper proves that the Jacobi equation is the unique non-trivial linear $k$-jet approximation to geodesic deviation equations invariant under certain coordinate transformations, with extensions to higher order deviations in Finsler geometry.

Contribution

It establishes the uniqueness of the Jacobi equation as the only linear $k$-jet approximation invariant under specific transformations and explores higher order deviations in Finsler geometry.

Findings

Jacobi equation is the only non-trivial linear $k$-jet approximation invariant under certain transformations.

Higher order geodesic deviation equations are studied for Finsler sprays.

Existence of invariant differential equations beyond linear approximations when linearity is not imposed.

Abstract

Let $M$ be a smooth manifold and $\mathcal{S}$ a semi-spray defined on a sub-bundle $\mathcal{C}$ of the tangent bundle $TM$. In this work it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of $\mathcal{S}$, linear on the deviation functions and invariant under an specific class of local coordinate transformations is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then 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 geodesic deviation equations for a Finsler spray.