1. CMB 2011 (vol 55 pp. 586)
|On Sha's Secondary Chern-Euler Class|
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 PoincarÃ©-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 PoincarÃ©-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 PoincarÃ©-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