# the slice theorem in the symplectic manifold

as we all know that the slice theorem is very important in symplectic geometry , especially in the proof of marsten-sternberg-weinstein reduction theorem . so I wonder a similar question that does there is a similar theorem when the symplectic manifold have some sigular point ? and i need a original proof of the slice theorem 'Sur certains groupes de transformations de Lie' by Koszul , if you have the electric version of this paper,please send a copy to me wangzhiwei08@gmail.com.

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## 2 Answers

I am only aware of results concerning singular symplectic reduction, i.e. when the Hamiltonian group action is not free and the quotient space is only a stratified symplectic space. The theorem is due to Sjamaar and Lerman ("Stratified symplectic spaces and reduction").

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Yeah , what you said is correct ,but what i want is in the not free case , ie the stratified symplectic space even clearly in the symplectic orbifold case ,does there exists such an analogue slice theorem ? –  HKSHLZW Nov 26 '10 at 9:25

I'm not sure if I understand your question correctly, but

Isenberg, J. & Marsden, J. E. A slice theorem for the space of solutions of Einstein's equations Physics Reports , 1982, 89, 179-222

extends the notion of a slice to the case where the underlying space is not a manifold but only a stratified space. They also proof a slice theorem in this setting, but this is clearly adjusted to the discussed case of General Relativity.

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