# A Remark on the Localization formulas about two Killing vector fields

## TL;DR

This paper explores localization formulas in equivariant cohomology involving two Killing vector fields, deriving new formulas for characteristic numbers and a Duistermaat-Heckman type formula on symplectic manifolds.

## Contribution

It introduces new localization formulas for two Killing vector fields and applies them to characteristic numbers and symplectic geometry.

## Key findings

- Derived localization formulas involving two Killing vector fields.
- Obtained formulas for characteristic numbers.
- Established a Duistermaat-Heckman type formula on symplectic manifolds.

## Abstract

In this article, we will discuss a localization formulas of equivariant cohomology about two Killing vector fields on the set of zero points ${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get formulas about characteristic numbers and to get a Duistermaat-Heckman type formula on symplectic manifold.