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

$$f(\lambda_1 x_1+\lambda_2 x_2,y)=\lambda_1 f(x_1,y)+\lambda_2 f(x_2,y),$$

$$f(y,x)=-f(x,y),$$

and

$$f(x,x)=0.$$

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