分级流形上齐次Pfaffian形式的Darboux分类
A Darboux classification of homogeneous Pfaffian forms on graded manifolds
AI总结：
研究超流形上关于给定齐次结构齐次的微分Pfaffian形式的局部分类问题，证明存在齐次Darboux坐标，从而得到齐次Pfaffian形式的Darboux型标准形。
摘要：
我们研究超流形 $M$ 上关于 $M$ 上给定齐次结构齐次的微分Pfaffian形式的局部分类问题。齐次结构最常见的例子是与向量丛结构相关的那些。我们的目标是证明，对于固定次数的齐次形式，存在齐次Darboux坐标。作为推论，我们得到齐次Pfaffian形式的Darboux型标准形，恢复经典Darboux定理及其接触和预辛对应作为特例。为了在超几何背景下表述Darboux分类的类似物，我们将特征分布 $\chi(\alpha)=\ker(\alpha)\cap\ker(\mathrm{d}\alpha)$ 关联到微分形式 $\alpha$，并将 $\alpha$ 的类定义为 $\chi(\alpha)$ 的秩。我们证明，在适当的正则性和常秩假设下，该分布完全决定了齐次Pfaffian形式的局部等价问题。我们的结果同样适用于普通（纯偶数）流形。
英文摘要：
We study the local classification problem for differential Pfaffian forms on a supermanifold $M$ that are homogeneous with respect to a given homogeneity structure on $M$. The most familiar examples of homogeneity structures are those associated with vector bundle structures. Our aim is to show that, for a homogeneous form of fixed degree, there exist homogeneous Darboux coordinates. As a consequence, we obtain Darboux-type normal forms for homogeneous Pfaffian forms, recovering as special cases the classical Darboux theorem together with its contact and presymplectic counterparts. To formulate an analogue of Darboux classification in the supergeometric setting, we associate to a differential form $α$ the characteristic distribution $χ(α)=\ker(α)\cap\ker(\mathrm{d}α)$, and define the class of $α$ as the rank of $χ(α)$. We prove that, under suitable regularity and constant-rank assumptions, this distribution completely determines the local equivalence problem for homogeneous Pfaffian forms. Our results apply equally well to ordinary (purely even) manifolds.
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