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# On Sha's Secondary Chern-Euler Class

Published:2011-05-13
Printed: Sep 2012
• Zhaohu Nie,
Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA
 Format: LaTeX MathJax PDF

## Abstract

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: transgression, secondary Chern-Euler class, locally product metric, law of vector fields
 MSC Classifications: 57R20 - Characteristic classes and numbers 57R25 - Vector fields, frame fields

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