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