On Sha's Secondary Chern-Euler Class

http://dx.doi.org/10.4153/CMB-2011-089-1
Canad. Math. Bull. 55(2012), 586-596

  Published:2011-05-13
 Printed: Sep 2012

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.