a *unit vector $x$. One also remarks that the standard structure corresponds to the north pole of $S^2$ and structures tamed by $\omega\st $ constitute the upper half-sphere.*

Exercise. 1. Note that $\jj = \{ J\in O(4): J^2=-E\} = \{ J: 
J^t=J^{-1}=-J\} $. Prove that condition $J^t=-J$ implies that $$J = \left(
\matrix
0&-s &c_1&c_2\cr
s &0&c_3&c_4\cr
-c_1&-c_3&0&-t \cr
-c_2&-c_4&t &0\cr
\endmatrix \right),$$ for some $s,t,c_i, i=1,...,4.$

[**2.**]{} Denote by $C$ the matrix $\matrix c_1& c_2 \cr c_3& c_4\endmatrix$. Prove that condition $J^2=-E$ implies that:

a\) $$C = \left(
\matrix
\pm c_1&c_2\cr
\mp c_2&c_1\cr
\endmatrix \right)$$ with $c_1^2+c_2^2=1-s^2$;

b\) $\vert t\vert =\vert s\vert $;

c\) and, finally, that $s=t$ and for $s\not= \pm 1$ the matrix ${1\over \sqrt{1-s^2}}\cdot C$ lies in $O_-(2)$ (the set of orthogonal $2\times 2$-matrices with determinant $-1$) iff the structure $$J_s := 
\left(
\matrix
0&-s &c_1&c_2\cr
s &0&c_3&c_4\cr
-c_1&-c_3&0&-s \cr
-c_2&-c_4&s &0\cr
\endmatrix \right)$$ defines the same orientation of $\rr^4$ as $J\st = J_1$.

[**3.**]{} Check finally that $\omega\st (x,J_sx)=s\cdot \Vert x\Vert^2$.

Let $M_1$ and $M_2$ be two smooth oriented surfaces in the unit ball $B\subset
\rr^4$ with zero as a common point. Let $v_1, w_1$ and $v_2, w_2$ be oriented bases of $T_0M_1$ and $T_0M_2$, 