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We consider the action of $S^{1}$ on $S^{2}$ by rotation respect the vertical axes. We want to integrate the $2-$ equivariant form

$$\alpha(X)= -X\cos(\phi) +\sin(\phi)\,d\phi\,d\theta.$$

We have the localization formula: $$\int_{S^{2}} \alpha(X)= (-2\pi)^{\ell} \sum_{p \in M_{0}(X)} \frac{\alpha(X)(p)}{\det(L_{p})^{1/2}}$$

(where $\ell=\dim(M)/2$, $\alpha(X)(p)$ the value of the function $\alpha(X)_{[0]}$ in the point $p$, $M_{0}(X)$ is the set of zeros of $X$ and for $p \in M_{0}(X)$ $L_{p}$ is a linear automorphism of $T_{p}(M)$ induced by the Lie action $L(X)\xi = [X, \xi]$ ). So we consider $X \in Lie(S^{1})= \mathbb{R}$. How can I explicity integrate this form ($\int_{S^{2}}\alpha(X)$)? (I know that the result is $2 \pi$)

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When we integrate an equivariant form on an $n$-manifold, we can ignore all terms except $n$-forms. (An $k$-form means the ordinary (non-equivariant) differential $k$-form.) In your case, you just integrate only the second term. – H. Shindoh Feb 1 '13 at 20:06
Same question on MO – Martin Feb 2 '13 at 8:40

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