标题： 分级喷射几何 显示英文标题

标题： Graded Jet Geometry

摘要： 射流流形和向量丛使得可以运用微分几何的工具来研究微分方程，例如在物理学中作为运动方程出现的方程。它们对于拉格朗日力学和变分法的几何表述是必要的。因此，在$\mathbb{Z}$-分次流形和向量丛的几何中，要求它们的推广是自然的。我们的目标是构造任意$\mathbb{Z}$-分次向量丛$\mathcal{E}$在任意$\mathbb{Z}$-分次流形$\mathcal{M}$上的$k$阶射流丛$\mathfrak{J}^{k}_{\mathcal{E}}$。我们通过直接构造其截面层来实现这一点，这使得能够快速证明其所有通常的性质。 结果发现，从构造$k$阶（线性）微分算子$\mathfrak{D}^{k}_{\mathcal{E}}$在$\mathcal{E}$上的分次向量丛开始是方便的。 在此过程中，我们讨论了（主）符号映射以及一类微分算子，其符号对应于完全对称的$k$向量场，从而找到了Atiyah李代数的分次版本。 回顾了在$\mathbb{Z}$分次流形上的$\mathbb{Z}$分次向量丛的几何学的基本知识。 显示英文摘要

摘要： Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of Lagrangian mechanics and the calculus of variations. It is thus only natural to require their generalization in geometry of $\mathbb{Z}$-graded manifolds and vector bundles. Our aim is to construct the $k$-th order jet bundle $\mathfrak{J}^{k}_{\mathcal{E}}$ of an arbitrary $\mathbb{Z}$-graded vector bundle $\mathcal{E}$ over an arbitrary $\mathbb{Z}$-graded manifold $\mathcal{M}$. We do so by directly constructing its sheaf of sections, which allows one to quickly prove all its usual properties. It turns out that it is convenient to start with the construction of the graded vector bundle of $k$-th order (linear) differential operators $\mathfrak{D}^{k}_{\mathcal{E}}$ on $\mathcal{E}$. In the process, we discuss (principal) symbol maps and a subclass of differential operators whose symbols correspond to completely symmetric $k$-vector fields, thus finding a graded version of Atiyah Lie algebroid. Necessary rudiments of geometry of $\mathbb{Z}$-graded vector bundles over $\mathbb{Z}$-graded manifolds are recalled.