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Geometry is the study of figures in a space of a given number of dimensions and of a given type. The most common types of geometry are plane geometry (dealing with objects ...

For every even dimension 2n, the symplectic group Sp(2n) is the group of 2n×2n matrices which preserve a nondegenerate antisymmetric bilinear form omega, i.e., a symplectic ...

A pair (M,omega), where M is a manifold and omega is a symplectic form on M. The phase space R^(2n)=R^n×R^n is a symplectic manifold. Near every point on a symplectic ...

A symplectic form on a smooth manifold M is a smooth closed 2-form omega on M which is nondegenerate such that at every point m, the alternating bilinear form omega_m on the ...

Informally, a symplectic map is a map which preserves the sum of areas projected onto the set of (p_i,q_i) planes. It is the generalization of an area-preserving map. ...

A map T:(M_1,omega_1)->(M_2,omega_2) between the symplectic manifolds (M_1,omega_1) and (M_2,omega_2) which is a diffeomorphism and T^*(omega_2)=omega_1, where T^* is the ...

A real-linear vector space H equipped with a symplectic form s.

The projective symplectic group PSp_n(q) is the group obtained from the symplectic group Sp_n(q) on factoring by the scalar matrices contained in that group. PSp_(2m)(q) is ...

A geometry constructed without reference to measurement. The only primitive concepts are those of points and intermediacy. There are 10 axioms underlying ordered geometry.