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Classical mechanics can be formulated in general spaces in terms of a Poisson bracket of functions {f,g}. The abstract properties of such a bracket lead to the study of Poisson manifolds and their symmetries. A special case is where the bracket is non-degenerate and then it comes from a closed non-degenerate 2-form w. A pair (M,w) consisting of a manifold and such a 2-form is called a symplectic manifold.
The geometry of symplectic manifolds is quite different from
Riemannian geometry but shares some features with complex manifolds,
especially Kaehler manifolds (which are a special case since the Kaehler
form is closed and non-degenerate).
Symplectic manfiolds can be studied from many points of view. Their
topology, differential geometry, symmetries have all been the object of
much study in recent times.
Differential geometry, quantisation, symmetry and momentum maps
(Rawnsley)
Kaehler manifolds
(Micallef)
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