Whitney公式与解析微分公式的基本关系
Formes de Whitney et primitives relatives de formes différentielles sous-analytiques
摘要：
设$X$是一个实解析流形，$g\colon X\to{\mathbf R}^n$是一个真可三角化的次解析映射.给定$X$上的一个亚解析$r$ -形式$ω$，其对$g$的每一个非奇异纤维的拉回都是精确的，我们证明了$ω$有一个相对本原：存在一个亚解析$(r-1)$ -形式$Ω$使得$dgΛ(ω-dΩ)=0$ .证明使用亚解析三角测量来将问题转化为与棱镜相关的“相对惠特尼形式”。利用Whitney形式的组合学，我们证明了这一结果最终是由一类特殊的线性偏微分方程解的次分析性得出的.这部作品的灵感来自于弗朗索瓦·特里维斯的一个问题。
英文摘要：
Let $X$ be a real-analytic manifold and $g\colon X\to{\mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $ω$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $ω$ has a relative primitive: there is a subanalytic $(r-1)$-form $Ω$ such that $dgΛ(ω-dΩ)=0$. The proof uses a subanalytic triangulation to translate the problem in terms of "relative Whitney forms" associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of François Treves.
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