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Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics. (English) Zbl 0933.58010

Let (M,·,·) be a closed oriented Riemannian manifold and 𝒟 μ s (M) the group of volume preserving diffeomorphisms of M. The weak L 2 right invariant Riemannian metric on 𝒟 μ s (M) is given by

X,Y L 2 = M X(x),Y(x)μ(x),

where X,YT e 𝒟 μ s (M) are vector fields on M and μ is the volume element on M. V. I. Arnold, D. Ebin and J. Marsden showed that geodesics η t of the weak L 2 right invariant Riemannian metric on 𝒟 μ s (M) 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

V ˙(t)+ U(t) V(t)-α 2 [U(t)] t ·ΔU(t)=-gradp(t)

where V=(1-α 2 Δ)U, divU=0.

In this paper, the equation of mean motion of an ideal fluid is generalized to the case of a manifold M. This model corresponds to the weak H 1 right invariant Riemannian metric on 𝒟 μ s (M) which is expressed as

X,Y 1 =X,(1+Ric)Y L 2 +X,Y L 2

for X,YT e 𝒟 μ s (M). The following problems are investigated: H 1 -covariant derivative and its geodesic flow on 𝒟 μ s (M), curvature of the H 1 -metric, existence and uniqueness results for the Jacobi equation, stability and curvature.

MSC:
58D05Groups of diffeomorphisms and homeomorphisms as manifolds
76M30Variational methods (fluid mechanics)
58B20Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds