测地偏差方程的$k$ -喷流场近似
On $k$-jet field approximations to geodesic deviation equations
摘要：
设$M$是光滑流形，$\mathcal{S}$是定义在切丛$TM$的子丛$\mathcal{C}$上的半喷集.在这项工作中，它被证明是唯一的非平凡的$k$ -喷气近似的精确测地偏差方程的$\mathcal{S}$，线性的偏差函数和不变的一类特定的局部坐标变换下的Jacobi方程。然而，如果不对偏差函数的依赖性施加线性性质，则存在微分方程，其解允许$k$ -jet近似并且在任意坐标变换下不变。作为高阶测地线偏差方程的一个例子，我们研究了Finsler喷雾的一阶和二阶测地线偏差方程。
英文摘要：
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.
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