On Sha's secondary Chern-Euler class

TL;DR

This paper investigates the secondary Chern-Euler class for manifolds with boundary, providing explicit transgression forms and connecting Sha's relative Poincaré-Hopf theorem to classical vector field indices.

Contribution

It explicitly constructs a transgression form showing the secondary Chern-Euler form is exact away from boundary normals, linking Sha's theorem to classical vector field indices.

Findings

Explicit transgression form for secondary Chern-Euler class

Boundary term evaluated via classical vector field indices

Sha's relative Poincaré-Hopf theorem is equivalent to classical law

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 used by Sha to formulate a relative Poincar\'e-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 Poincar\'e-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 Poincar\'e-Hopf theorem is equivalent to the more classical Law of Vector Fields.