A Poincaré-Hopf Type Formula for A Pair of Vector Fields

2018/01/08 by Xu Chen, Chen, Xu
Mathematics · #53C20 #53C99 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1801.02487

openalex publication_date 2018/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For two complex vector bundles admitting a homomorphism between them, a Poincaré-Hopf formula for the difference of the Chern character numbers of these two vector bundles with isolated singularities is established by Huitao Feng, Weiping Li and Weiping Zhang. This article extend their reslut about Poincaré-Hopf type formula for the difference of the Chern character numbers to the non-isolated singularities. A special connection \widetilde∇₁^E be constructed, the Chern character \rm ch(E,\widetilde∇₁^E) is the key role for the non-isolated singularities. As a consequence, a Poincaré-Hopf type formula for a pair of vector field with the function h^T_ℂM(⋅,⋅) has non-isolated zero points over a closed, oriented smooth manifold of dimension 2n is established.

Related

• A Poincaré-Hopf type formula for Chern character numbers
• Relative Chern character number and super-connection
• On Sha's secondary Chern-Euler class
• Explicit computation of the Chern character forms
• Secondary Chern-Euler forms and the Law of Vector Fields