Sur la systole de la sphère au voisinage de la métrique standard

We study the systolic area (defined as the ratio of the area over the square of the systole) of the $2$-sphere endowed with a smooth riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, have the standard metric $g_0$ for critic point, although this one do not achieve the conjectured global minimum : we show that for each tangent direction to the space of metrics at $g_0$, there exists a variation by metrics corresponding to this direction along which the systolic area can only increase