On Sha's secondary Chern-Euler class - Mathematics > Differential GeometryReport as inadecuate

On Sha's secondary Chern-Euler class - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: For a manifold with boundary, the restriction of Chern-s transgression formof the Euler curvature form over the boundary is closed. Its cohomology classis called the secondary Chern-Euler class and used by Sha to formulate arelative Poincar\-e-Hopf theorem, under the condition that the metric on themanifold is locally product near the boundary. We show that the secondaryChern-Euler form is exact away from the outward and inward unit normal vectorsof the boundary by explicitly constructing a transgression form. Using Stokes-theorem, this evaluates the boundary term in Sha-s relative Poincar\-e-Hopftheorem in terms of more classical indices of the tangential projection of avector field. This evaluation in particular shows that Sha-s relativePoincar\-e-Hopf theorem is equivalent to the more classical Law of VectorFields.

Author: Zhaohu Nie

Source: https://arxiv.org/

Related documents