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 D̂μ(S ) of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with the L-metric, whose geodesics correspond solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misio lek 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.