that the Fubini-Study metric induces on $L$ the standard spherical metric ${{\textsf s }}$ with area $1$. We assume with no loss of generality that $z$ and $w$ are the North and South pole of $L$. Let $E$ be the equator of $L$. For any $y \in E$, we denote by $M_y$ the meridian containing $y$, and by $M_y \{ \delta \}$ the $\delta$-neighbourhood of $M_y$ in $L$ for the spherical metric. The constants $0 < \theta < \theta'
< 2\zeta$ have been defined in subsection \[cons\]. Now we let $0 < \zeta < \zeta' < \zeta'' < 1$ satisfying: $$\label{sop}
\theta' < \zeta'' - \zeta'   \ \textrm{ and } \  \theta + \zeta'' < 1.$$ We may take for $(\theta,\theta',\zeta, \zeta' ,\zeta'')$ suitable multiples of a small $\theta > 0$. The second inequality of (\[sop\]) will be used in next subsection. The integer $n_0$ has been defined in subsection \[cons\].

\[touch\] Let $(z,w)$ be distinct points in ${\mathbb {P}}^k \setminus (\DD \cup \VV )$. With the above notations, there exists a subset $F \subset E$ of full Lebesgue measure satisfying the following properties. For any $y \in F$, there exists $n_{z,w}(y) \geq n_0$ such that:

1.  the meridian $M_y$ 