symplectic structure

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• Symplectic geometry — is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2 form. Symplectic geometry has its origins in the Hamiltonian formulation of classical… …   Wikipedia

• Symplectic manifold — In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2 form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology.… …   Wikipedia

• Symplectic filling — In mathematics, a filling of a manifold X is a cobordism W between X and the empty set. More to the point, the n dimensional topological manifold X is the boundary of an n+1 dimensional manifold W . Perhaps the most active area of current… …   Wikipedia

• Symplectic sum — In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a… …   Wikipedia

• Symplectic cut — In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that… …   Wikipedia

• Symplectic vector field — In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if (M,omega) is a symplectic manifold, then a vector field Xinmathfrak{X}(M) is symplectic if its flow preserves the symplectic… …   Wikipedia

• Symplectic matrix — In mathematics, a symplectic matrix is a 2n times; 2n matrix M (whose entries are typically either real or complex) satisfying the condition:M^T Omega M = Omega,.where MT denotes the transpose of M and Omega; is a fixed nonsingular, skew… …   Wikipedia

• Symplectic vector space — In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew symmetric, bilinear form omega; called the symplectic form. Explicitly, a symplectic form is a bilinear form omega; : V times; V rarr; R which is *… …   Wikipedia

• Symplectic group — For finite groups with all characteristc abelian subgroups cyclic, see group of symplectic type. Group theory …   Wikipedia

• Symplectic representation — In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space ( V , omega; ) which preserves the symplectic form omega; . Here omega; is a nondegenerate… …   Wikipedia

• Generalized complex structure — In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures… …   Wikipedia