次陈-欧拉形式与向量场定律
Secondary Chern-Euler forms and the Law of Vector Fields
摘要：
向量场定律是戈特利布为一个相对的庞加莱-霍普夫定理创造的术语。它首先由Morse证明，并用一般向量场的指数及其在边界上切向投影的内部来表示带边界带边流形的欧拉示性数.我们给出了这个拓扑定理的两个微分几何证明，其中二次陈-欧拉形式自然发挥了重要作用。在第一个证明中，要点是构造一个远离一些奇点的链。第二个证明采用了一个详细的研究第二陈-欧拉形式的边界上，这可能是独立的利益。更确切地说，我们通过显式地构造一个基元来证明，远离向外和向内的单位法向量，次陈-欧拉形式精确到拉回形式。应该强调的是，我们在一般情况下得到这个结果，而不假设度量是边界附近的局部乘积。在任何一种情况下，斯托克斯定理都被用来完成证明。
英文摘要：
The Law of Vector Fields is a term coined by Gottlieb for a relative Poincaré-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two differential-geometric proofs of this topological theorem, in which secondary Chern-Euler forms naturally play an essential role. In the first proof, the main point is to construct a chain away from some singularities. The second proof employs a detailed study of the secondary Chern-Euler form on the boundary, which may be of independent interest. More precisely, we show by explicitly constructing a primitive that, away from the outward and inward unit normal vectors, the secondary Chern-Euler form is exact up to a pullback form. It should be emphasized that we obtain this result in the general case without assuming the metric is locally product near the boundary. In either case, Stokes' theorem is used to complete the proof.
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