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:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)