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Is there some reference containing an ellaboration of Morse Theory for proper actions of Lie groups?

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Have a look at

N. Hingston, Equivariant Morse theory and closed geodesics, J. Differential Geom. Volume 19, Number 1 (1984), 1-276.

in particular Section 2. Most statements from the equivariant Morse theory for compact group actions go over unchanged, thanks to the existence of slices for proper actions; see

R. Palais, On the existence of slices for actions of noncompact Lie groups, Ann. of Math. 73 (1961)295-323.

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