## 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.## References

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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
- MR Author ID: 670293
- Email: znie@psu.edu
- Received by editor(s): December 15, 2010
- Published electronically: July 1, 2011
- Communicated by: Jianguo Cao
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**140**(2012), 1085-1096 - MSC (2000): Primary 57R20, 57R25
- DOI: https://doi.org/10.1090/S0002-9939-2011-11214-0
- MathSciNet review: 2869093