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Secondary Chern-Euler forms and the law of vector fields

Author: Zhaohu Nie
Journal: Proc. Amer. Math. Soc. 140 (2012), 1085-1096
MSC (2000): Primary 57R20, 57R25
Published electronically: July 1, 2011
MathSciNet review: 2869093
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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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Additional Information

Zhaohu Nie
Affiliation: Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601
Address at time of publication: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322

Received by editor(s): December 15, 2010
Published electronically: July 1, 2011
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society