Local Indices of a Vector Field at an Isolated Zero on the Boundary

TL;DR

This paper introduces two local index definitions for vector fields at boundary zeros and proves Poincare-Hopf-type theorems for manifolds with such vector fields, extending classical index theory to boundary cases.

Contribution

It defines new local indices at boundary zeros and establishes related Poincare-Hopf theorems for vector fields on manifolds with boundary.

Findings

Defined two types of local indices at boundary zeros

Proved Poincare-Hopf-type theorems for these indices

Extended classical index theory to manifolds with boundary

Abstract

We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros.