L_infty，loc -形式的Poincaré引理

形式流形的庞加莱引理

Poincaré's lemma for formal manifolds

本文在形式流形与形式李群理论框架下，为系数取形式函数、形式广义函数、紧支撑形式密度或紧支撑形式分布的德拉姆复形建立了庞加莱引理。

几何Poincaré引理

Geometric Poincaré Lemma

对微分链的拓扑向量空间建立了Poincaré引理的一个几何形式。特别地，光滑n-流形M的可收缩开子集U中的每个具有紧支撑的微分k-圈是U中具有紧支撑的微分（k+1）-链的边界。应用包括推广的中值定理和罗尔定理。

Poincaré's lemma on some non-Euclidean structures

In this paper we prove the Poincaré lemma on some $n$-dimensional corank 1 sub-Riemannian structures, formulating the $\frac{(n-1)n(n^2+3n-2)}{8}$ necessarily and sufficiently 'curl-vanishing' compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. Our proof is based on a Poincaré lemma stated on Riemannian manifolds and a suitable Cesàro-Volterra path integral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic …

关于de-bar-sub-M II的Poincaré引理的失效

On the failure of the Poincaré Lemma for de-bar-sub-M II

我们得到了关于切向柯西-黎曼方程的Poincare引理无效的非常尖锐的结果，该方程作用在切向形式上，切向于一般CR维数n和一般CR余维数k的CR流形M.这推广了经典的H.路易我们还讨论了CR结构的特征丛M，由于某些退化的Levi形式。一些自然发生的几何例子。

The Poincare lemma, antiexact forms, and fermionic quantum harmonic oscillator

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator o…

An $\mathsf{A}_{\infty}$ version of the Poincaré lemma

We prove a categorified version of the Poincaré lemma. The natural setting for our result is that of $\infty$-local systems. More precisely, we show that any smooth homotopy between maps $f$ and $g$ induces an $\mathsf{A}_\infty$-natural transformation between the corresponding pullback functors. This transformation is explicitly defined in terms of Chen's iterated integrals. In particular, we show that a homotopy equivalence induces a quasi-equivalence on the DG categories of $\infty$-local system.