Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 88-124 (2003).

The author discusses Hamiltonian actions on symplectic manifolds and gives a self-contained introduction to the Cartan model of equivariant cohomology. He relates the results to the Cartan theorem asserting that the

$G$-equivariant cohomology algebra

${H}_{G}^{*}\left(M\right)$ is isomorphic to the de Rham cohomology with complex coefficients of the orbit manifold

$M/G$, where

$G$ is a compact connected Lie group acting smoothly and freely on a smooth manifold

$M$. Then the author proves the major result of the paper, the equivariant localization theorem about computing

${\int}_{M}\alpha \left(\xi \right)$ for any

$G$-equivariantly closed differential form

$\alpha $ on

$M$ and any nondegenerate element

$\xi \in \U0001d524$ for which the associated vector field

${\xi}^{\#}$ has only isolated zeros, where

$\U0001d524$ is the Lie algebra of a compact Lie group

$G$ acting smoothly on a compact oriented manifold

$M$ of dimension

$2k$. As an application of the theorem, the author derives the generalized Duistermaat-Heckman theorem about computing

${\int}_{M}{e}^{i\mu \left(\xi \right)}{\nu}_{\omega}$ for any compact symplectic manifold

$(M,\omega )$ of dimension

$2k$ with a Hamiltonian action of

$G$ and corresponding symplectic moments given by

$\mu :\U0001d524\to {C}^{\infty}\left(M\right)$, where

$M$ is oriented with the Liouville form

${\nu}_{\omega}=\frac{1}{k!}{\omega}^{k}$.