1. CMB 2011 (vol 55 pp. 586)
 Nie, Zhaohu

On Sha's Secondary ChernEuler 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 ChernEuler 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 ChernEuler 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 ChernEuler class, locally product metric, law of vector fields Categories:57R20, 57R25 

2. CMB 2011 (vol 55 pp. 368)
 Nie, Zhaohu

The Secondary ChernEuler Class for a General Submanifold
We define and study the secondary ChernEuler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with nonisolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.
Keywords:secondary ChernEuler class, normal sphere bundle, Euler characteristic, index, nonisolated singularities, blowup Category:57R20 
