Symplectic Manifold - Definition

Definition

A symplectic form on a manifold M is a closed non-degenerate differential 2-form ω. The non-degeneracy condition means that for all p ∈ M we have the property that there does not exist non-zero X ∈ T_pM such that ω(X,Y) = 0 for all Y ∈ T_pM. The skew-symmetric condition (inherent in the definition of differential 2-form) means that for all p ∈ M we have ω(X,Y) = −ω(Y,X) for all X,Y ∈ T_pM. Recall that in odd dimensions antisymmetric matrices are not invertible. Since ω is a differential two-form the skew-symmetric condition implies that M has even dimension. The closed condition means that the exterior derivative of ω, namely dω, is identically zero. A symplectic manifold consists a pair (M,ω), of a manifold M and a symplectic form ω. Assigning a symplectic form ω to a manifold M is referred to as giving M a symplectic structure.