In this paper the notions of vector field and differential form are extended to locally compact groups which are the inverse limit of Lie groups. This is done using Bruhat's definition of [unk]c∞ functions on such a group. Vector fields are defined as derivations on the [unk]c∞ functions. Then tangent vectors at a point are defined as elements of the inverse limit of the tangent spaces of the Lie groups. Tangent vectors then are put together to form vector fields, corresponding to a bundle definition, and the two notions are shown to be equivalent. Differential forms are defined using a bundle type definition from continuous linear functional on the tangent space. An existence and uniqueness theorem is proven for the exterior differential. Then an analog of the Poincaré lemma leads to the de Rham theorem relating the Cech cohomology with real coefficients to the cohomology of the differential forms.
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机译：在本文中，矢量场和微分形式的概念被扩展到局部紧群，这是李群的反极限。这是通过在这样的组上使用Bruhat对[unk] c ∞函数的定义来完成的。向量字段定义为[unk] c ∞函数的派生。然后，将点处的切向量定义为李群切线空间的逆极限的元素。然后将切线向量放在一起以形成向量域，该向量域对应于束定义，并且两个概念被显示为等效。使用束类型定义从切线空间上的连续线性泛函定义微分形式。证明了外部微分的存在性和唯一性定理。然后，庞加莱引理的一个类似物导致了De Rham定理，该定理将具有实系数的Cech同调与微分形式的同调相关。
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