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

TL;DR

This paper explores the geometric properties of a group related to the Euler equation with Coriolis force, demonstrating positive curvature which has implications for the stability and structure of fluid flows on the sphere.

Contribution

It provides the first calculation of the Misiolek curvature for this specific group, linking geometric positivity to fluid dynamics with Coriolis effects.

Findings

Misiolek curvature is positive for the group studied

Positivity of curvature implies the existence of conjugate points

Results suggest stability properties of the fluid flow on the sphere

Abstract

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.