Mathematics – Differential Geometry
Scientific paper
2009-01-17
Mathematics
Differential Geometry
9 pages. Improved exposition. Final version to appear in "Canadian Mathematical Bulletin"
Scientific paper
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.
No associations
LandOfFree
On Sha's secondary Chern-Euler class does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Sha's secondary Chern-Euler class, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Sha's secondary Chern-Euler class will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-118946