(On the systole of the sphere in the proximity of the standard metric)
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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, admits the standard metric g 0 as a critical point, although it does 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.
KeywordsCritical point 2-sphere Standard metric Systole Zoll metric
Sur la systole de la sphère au voisinage de la métrique standard
Mathematics Subject Classifications (2000)53C22 58E10 37C27
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