the restriction of the symplectic form on to is non-degenerate. This imparts a symplectic structure to in a natural way. Then the action of on is also Hamiltonian, with momentum map the composition of the inclusion map with 's momentum map. Symplectic quotients Suppose that the action of a Lie group G on the symplectic manifold (M, ω) is Hamiltonian, as defined above, with momentum map . From the Hamiltonian condition, it follows that is invariant under G. Assume now that G acts freely and properly on . It follows that 0 is a regular value of , so and its quotient are both smooth manifolds. The quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to equals the restriction of ω to . Thus, the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, after , symplectic quotient, or symplectic reduction of M by G and is denoted . Its dimension equals the dimension of M minus twice the dimension of G. More generally, if G does not act freely (but still properly), then showed that is a stratified symplectic space, i.e. a stratified space with compatible symplectic structures on the strata. Flat connections on a surface The space of connections on the trivial bundle on a surface carries an infinite dimensional symplectic form The gauge group acts on connections by conjugation . Identify via the integration pairing. Then the map that sends a connection to its curvature is a moment map for the action of the gauge group on connections. In particular the moduli space of flat connections modulo gauge equivalence is given by symplectic reduction. See also GIT quotient Quantization commutes with reduction. Poisson–Lie group Toric manifold Geometric Mechanics Kirwan map Kostant's convexity theorem Notes References J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970. . S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Science Publications, 1990. . Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, Oxford Science Publications, 1998. . Symplectic geometry Hamiltonian mechanics Group actions (mathematics)