Positivity for the curvature of the diffeomorphism group corresponding to the incompressible Euler equation with Coriolis force

Abstract

Abstract We investigate the geometry of the central extension $\widehat{\mathcal D}_{\mu}(S^{2})$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with an $L^{2}$-metric, for which geodesics correspond to solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misiołek 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.