Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Secondary Chern-Euler forms and the law of vector fields
HTML articles powered by AMS MathViewer

by Zhaohu Nie PDF
Proc. Amer. Math. Soc. 140 (2012), 1085-1096 Request permission


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.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57R20, 57R25
  • Retrieve articles in all journals with MSC (2000): 57R20, 57R25
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:
  • 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:
  • MathSciNet review: 2869093