Secondary Chern-Euler forms and the law of vector fields

Nie, Zhaohu

doi:10.1090/S0002-9939-2011-11214-0

Secondary Chern-Euler forms and the law of vector fields
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Proc. Amer. Math. Soc. 140 (2012), 1085-1096 Request permission

Abstract:

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 elementary 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 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. In either case, Stokes’ theorem is used to complete the proof.

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