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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

\begin{displaymath}
f(\lambda_1 x_1+\lambda_2 x_2,y)=\lambda_1 f(x_1,y)+\lambda_2 f(x_2,y),
\end{displaymath}


\begin{displaymath}
f(y,x)=-f(x,y),
\end{displaymath}

and

\begin{displaymath}
f(x,x)=0.
\end{displaymath}

Symplectic forms can exist on $M$ (or $V$) only if $M$ (or $V$) is Even-dimensional.




© 1996-9 Eric W. Weisstein
1999-05-26