[*For any $g$-orthonormal system $v_1,...,v_{2k}$ in $\rr^{2n}$ one has $$\vert \sigma_k(v_1,...,v_{2k})\vert \le 1, \eqno(1.3.1)$$ with equality taking place iff the subspace $\<v_1,...,v_{2k}\>$ is $J$ - invariant.* ]{}

Remark. We shall also call $J$-invariant subspaces $J$-complex.

Proof. We shall prove (1.3.1) by induction on $k$.

Let $k=1$. Then $\vert \omega (v_1,v_2)\vert =\vert g(v_1,Jv_2)\vert
\le \Vert v_1\Vert_g\cdot \Vert Jv_2\Vert_g=$ $\Vert v_1\Vert_g\cdot \Vert v_2
\Vert_g$, with equality iff $v_1$ and $Jv_2$ are collinear. Thus the subspace $\<v_1,v_2\>$ is $J$-invariant.

Now let $V$ be a subspace of dimension $2k$. Put $\omega'=\omega\mid_V$. Find a $g$-orthonormal base $e_1,...,e_{2k}$ of $V$, s.t. $\omega'=
\lambda_1e^1\wedge e^2 + ... +\lambda_ke^{2k-1}\wedge e^{2k}$. By the case $k=1$ we have $\vert \omega' (e_{2p-1},e_{2p})\vert=
\vert \omega (e_{2p-1},e_{2p})\vert =\vert \lambda_p\vert \le 1$, with equality taking place iff $e_{2p}={+\over -}Je_{2p-1}$. So, for $\sigma'_k
=\sigma_k\mid_V$ we obtain $$\vert \sigma_k'(e_1,...,e_{2k})\vert = \vert {1\over k!}\omega^k(e_1,...,
e_{2k})\vert = \vert \lambda_1...\lambda_k\vert \le 1,$$ with equality iff $e_{2p}={+\over -}Je_{2p-1}$ for all $1\le p\le k$.

A submanifold $Y$ of an almost-complex manifold $(X,J)$ is called an almost-complex (or $J$-complex) if the tangent spaces to $Y$ are $J$-invariant.

Definition 1.3.1. [*Nowhere degenerate exterior two-form $\omega$ on $X$ is called symplectic if $d\omega =0$.*]{}

A pair $(X,\omega )$ is called a symplectic manifold.

Corollary 