Some positivity results of the curvature on the group corresponding to the incompressible Euler equation with Coriolis force

Abstract

In this article, we investigate the geometry of a central extension ${\hat{\mathcal{D}}}_{\mu }(S^2)$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with the $L^2$-metric, whose geodesics correspond solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misiolek curvature of this group. This value is related to the existence of a conjugate point and its positivity directly implies the positivity of the sectional curvature.

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