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

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