## Duistermaat-Heckman Formula and Bott’s Original Idea

1. Duistermaat-Heckman Formula In this section, we consider the case of that $\left(M^{2l},\omega\right)$ is a symplectic manifold. Let $(M,\omega)$ be a symplectic manifold with $\omega$ is a symplectic structure. It means

1. $\omega$ is a non-singular 2-form. i.e. If for any $Y\in \Gamma(TM)$ there always have $\omega(X,Y)=0$, then $X=0$.
2. $\rd \omega=0$.
We make the same assumptions as in previous section. Let $i_1,\cdots,i_k$ be $k$ positive even integers. For any $p\in\mathrm{zero}(K)$ and $1\leq j \leq k$, set
$\lambda^{i_j}(p)=\lambda_1^{i_j}+\cdots+\lambda_l^{i_j}.$By following theorem, we reduce the computation of characteristic numbers of $TM$ to quantities on $\mathrm{zero}(K)$.