Let be a closed oriented Riemannian manifold and the group of volume preserving diffeomorphisms of . The weak right invariant Riemannian metric on is given by
where are vector fields on and is the volume element on . V. I. Arnold, D. Ebin and J. Marsden showed that geodesics of the weak right invariant Riemannian metric on are motions of incompressible ideal fluids. D. D. Holm, J. E. Marsden and T. S. Ratiu [Adv. Math. 137, 1-81 (1998)] derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation
where , .
In this paper, the equation of mean motion of an ideal fluid is generalized to the case of a manifold . This model corresponds to the weak right invariant Riemannian metric on which is expressed as
for . The following problems are investigated: -covariant derivative and its geodesic flow on , curvature of the -metric, existence and uniqueness results for the Jacobi equation, stability and curvature.