关于ℤ-分次几何中分布的可积性
On the Integrability of Distributions in $\mathbb{Z}$-graded Geometry
AI总结：
本文研究ℤ-分次流形上分布的可积性，证明局部弗罗贝尼乌斯定理，讨论两种积分子流形概念，明确整体弗罗贝尼乌斯定理的单向蕴含关系，给出反向蕴含的反例。
AI中文摘要：
本文研究ℤ-分次流形上分布的可积性，首先证明了对合分布的平坦坐标局部存在性——即局部弗罗贝尼乌斯定理，随后讨论两种不同的积分子流形概念，发现对其中较强的概念，整体弗罗贝尼乌斯定理仅成立单向蕴含：对合性蕴含可积性，还证明可积性蕴含某一较弱版本的对合性，并给出了反向蕴含的简单反例。
英文摘要：
Integrability of distributions on $\mathbb{Z}$-graded manifolds is examined. First, the Local Frobenius theorem -- the local existence of flat coordinates for an involutive distribution -- is proved. Then, two different notions of an integral submanifold are discussed. It is found that, for the stronger of the two notions, the Global Frobenius theorem holds, but as one implication only: involutivity implies integrability. It is then shown that integrability implies a certain weaker version of involutivity. Simple counterexamples of the converse implications are given.
