Riemannian curvature on the group of area-preserving diffeomorphisms (motions of fluid) of 2-sphere

Abstract

Euler's equation for an incompressible fluid filled in a Riemannian manifold D is regarded as a geodesic equation on the group of volume-preserving diffeomorphisms of D provided with a one-sided invariant metric. A negative sectional curvature implies instability of the geodesic with respect to the corresponding flow and perturbation. The exponential growth of the perturbation is estimated from the values of the sectional curvatures. This paper presents the expression of the components of Riemannian curvature tensor of the group of area-preserving diffeomorphisms of a 2-sphere in explicit formulas through 3 − j coefficients.