Symplectic Form

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 tangent space T_mM is nondegenerate.

A symplectic form on a vector space V over F_q is a function f(x,y) (defined for all x,y in V and taking values in F_q) which satisfies




f is called non-degenerate if f(x,y)=0 for all y implies that x=0. Symplectic forms can exist on M (or V) only if M (or V) is even-dimensional. An example of a symplectic form over a vector space is the complex Hilbert space with inner product <·,·> given by


See also

Symplectic Space, Vector Space

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Cite this as:

Weisstein, Eric W. "Symplectic Form." From MathWorld--A Wolfram Web Resource.

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