+ dx_2\wedge dy_2 $ and the standard complex structure $J\st$ defined by the operator $$J\st =
\left( \matrix
0&-1&0&0\cr
1&0&0&0\cr
0&0&0&-1\cr
0&0&1&0\cr
\endmatrix \right).$$

Let $\jj$ denote the set of all orthogonal complex structures in $\rr^4$ giving $\rr^4$ the same orientation as $J\st$. Orthogonality here means just that $J$ is an orthogonal matrix. The same orientaion means that for any pair $x_1, x_2$ of $J$-independent vectors the basis $x_1, Jx_1, x_2, Jx_2$ gives the same orientation of $\rr^4$ as $J\st$.

Exercise. [Prove that this orientation does not depend on the particular choice of $x_1, x_2$ and coincides with the orientation given by $\omega\st^2$. ]{}

The following lemma summarizes the elementary facts which we need for the sequel.

Lemma 1.4.1. *The elements of $\jj$ have the form $$J_{s} =
\left(
\matrix
0&-s &c_1&c_2\cr
s &0&c_2&-c_1\cr
-c_1&-c_2&0&-s \cr
-c_2&c_1&s &0\cr
\endmatrix \right),\eqno(1.4.1)$$ with $c_1^2 + c_2^2 + s^2 = 1$. One also has for $x\in \rr^4$ $$\omega\st  (x, J_s x) = s \Vert x\Vert^2 . \eqno(1.4.2)$$*

We remark that the set $\jj$ is a unit two-dimensional sphere $S^2$ in $\rr^3$ with coordinates $c_1, c_2,s $. We note also that the number $\omega\st (x, J_s x)$ does not depend on the choice of 