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

In this article, we investigate the geometry of a central extension $\widehat{\mathcal D}_μ(S^{2})$ 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.