PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Aissa Wade.

Title:Secondary Chern-Euler forms and the Law of Vector Fields
Seminar:Topology/Geometry Seminar
Speaker:Zhaohu Nie, Penn State
Abstract:
In this talk, I will present results generalizing those in my previous related talks. The improvement comes from removing the restrictive condition that the metric on the manifold is locally product near its boundary. The outline of the talk is as follows. In his famous proof of the Gauss-Bonnet theorem, Chern transgressed the Euler curvature form of a Riemannian manifold on the tangent sphere bundle. For a manifold with boundary, the restriction of Chern's transgression form to the tangent sphere bundle on the boundary is closed, and we call it the secondary Chern-Euler form. In this talk, we will show that this form is exact, up to a pullback form, away from the inward and outward unit normal vectors of the boundary by explicitly constructing a primitive. I should emphasize again that we can do this for any metric on the manifold, not necessary locally product near the boundary. Using Stokes' theorem, this then evaluates some boundary term and thus proves the Law of Vector Fields in a differential geometric fashion. Recall that the Law of Vector Fields is a relative version of the Poincar\'e-Hopf theorem and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. Some related results will also be mentioned.

Room Reservation Information

Room Number:MB106
Date:10 / 15 / 2009
Time:05:15pm - 06:30pm