Bott and Duistermaat-Herckman Formulas

In Chapter One we have defined characteristic classes and numbers. A natural question is hoe to compute these characteristic numbers. Let $\omega$ be a characteristic form on an even dimensional smooth closed oriented manifold $M$. If
\[
\omega=\omega_{[1]}+\omega_{[2]}+\cdots+\omega_{[\dim M]},\quad \omega_{[i]}\in\Omega^{i}(M), i=1,\cdots,\dim M,
\] then the characteristic number associated $\omega$ is defined by $\int_{M}\omega=\int_M\omega_{[\dim M]}$. The Bott’s result shows that
\[
\int_M\omega_{[\dim M]}=\sum_{p\in A}\mu(P),
\]
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