Natural operations in differential geometry
Natural operations in differential geometry
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DOI：
10.1007/978-3-662-02950-3
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
I. Kolář;J. Slovák;P. Michor
中科院分区：
文献类型：
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作者：
I. Kolář;J. Slovák;P. Michor
这项工作的目的有三个：首先，它应该是一部关于微分几何中自然丛和自然算子的专题著作。这是一个每个微分几何学家都曾多次遇到的领域，但没有在一个地方详细讨论过。让我们解释一下自然性的含义。外导数随着微分形式的回调而交换。本声明的背景是以下一般概念。向量丛 A kT* M 实际上是函子的值，它将 M 上的丛与每个流形 M 相关联，并将 f 上的向量丛同态与相同维度的流形之间的每个局部微分同态 f 相关联。这是自然束概念的一个简单示例。对于每个流形 M，外导数 d 将 A kT* M 的部分变换为 A k+ 1T* M 的部分，这一事实可以通过说 d 是从 A kT* M 到 A k+ 1T* M 的算子来表达。
The aim of this work is threefold: First it should be a monographical work on natural bundles and natural op erators in differential geometry. This is a field which every differential geometer has met several times, but which is not treated in detail in one place. Let us explain a little, what we mean by naturality. Exterior derivative commutes with the pullback of differential forms. In the background of this statement are the following general concepts. The vector bundle A kT* M is in fact the value of a functor, which associates a bundle over M to each manifold M and a vector bundle homomorphism over f to each local diffeomorphism f between manifolds of the same dimension. This is a simple example of the concept of a natural bundle. The fact that exterior derivative d transforms sections of A kT* M into sections of A k+ 1T* M for every manifold M can be expressed by saying that d is an operator from A kT* M into A k+ 1T* M.