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On Sha's Secondary Chern–Euler Class

Published online by Cambridge University Press:  20 November 2018

Zhaohu Nie
Zhaohu Nie*
Affiliation:
Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USAe-mail: [email protected]
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Abstract

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For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

Keywords

57R2057R25transgressionsecondary Chern–Euler classlocally product metriclaw of vector fields
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 3 , 01 September 2012 , pp. 586 - 596
Copyright
Copyright © Canadian Mathematical Society 2012

References

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[4] Morse, M., Singular points of vector fields under general boundary conditions. Amer. J. Math. 51(1929), no. 2, 165–178. http://dx.doi.org/10.2307/2370703 Google Scholar
[5] Nie, Z., Secondary Chern-Euler forms and the law of vector fields. arXiv:0909.4754.Google Scholar
[6] Sha, J.-P., A secondary Chern–Euler class. Ann. of Math. 150(1999), no. 3, 1151–1158. http://dx.doi.org/10.2307/121065 Google Scholar