A Poincar\'e-Hopf Type Formula for A Pair of Vector Fields

For two complex vector bundles admitting a homomorphism between them, a Poincar\'e-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 Poincar\'e-Hopf type formula for the difference of the Chern character numbers to the non-isolated singularities. A special connection $\widetilde{\nabla}_{1}^{E}$ be constructed, the Chern character ${\rm ch}(E,\widetilde{\nabla}_{1}^{E})$ is the key role for the non-isolated singularities. As a consequence, a Poincar\'e-Hopf type formula for a pair of vector field with the function $h^{T_{\mathbb{C}}M}(\cdot,\cdot)$ has non-isolated zero points over a closed, oriented smooth manifold of dimension $2n$ is established.