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 ∈ TpM such that ω(X,Y) = 0 for all Y ∈ TpM. 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 ∈ TpM. 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.
Read more about this topic: Symplectic Manifold
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)