On Sha's Secondary Chern-Euler Class

Canadian mathematical bulletin

Zhaohu Nie¹ 

[1] Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA

关键词: transgression;    secondary Chern-Euler class;    locally product metric;    law of vector fields;   

DOI  :  10.4153/CMB-2011-089-1

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

For a manifold with boundary, the restriction of Chern's transgressionform of the Euler curvature form over the boundary is closed. Itscohomology class is called the secondary Chern-Euler class and wasused by Sha to formulate a relative Poincaré-Hopf theorem under thecondition that the metric on the manifold is locally product near theboundary. We show that the secondary Chern-Euler form is exact awayfrom the outward and inward unit normal vectors of the boundary byexplicitly constructing a transgression form. Using Stokes' theorem,this evaluates the boundary term in Sha's relative Poincaré-Hopftheorem in terms of more classical indices of the tangentialprojection of a vector field. This evaluation in particular showsthat Sha's relative Poincaré-Hopf theorem is equivalent to the moreclassical law of vector fields.

【 授权许可】

Unknown