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Geometry and curvature of diffeomorphism groups with $$H^1$$ metric and mean hydrodynamics. (English) Zbl 0933.58010
Let $$(M,\langle \cdot ,\cdot \rangle)$$ be a closed oriented Riemannian manifold and $$\mathcal D_\mu ^s(M)$$ the group of volume preserving diffeomorphisms of $$M$$. The weak $$L^2$$ right invariant Riemannian metric on $$\mathcal D_\mu ^s(M)$$ is given by $\langle X,Y\rangle _{L^{2}}=\int_{M}\langle X(x),Y(x)\rangle \mu (x),$ where $$X,Y\in T_e\mathcal D_\mu ^s(M)$$ are vector fields on $$M$$ and $$\mu$$ is the volume element on $$M$$. V. I. Arnold, D. Ebin and J. Marsden showed that geodesics $$\eta _t$$ of the weak $$L^2$$ right invariant Riemannian metric on $$\mathcal D_\mu ^s(M)$$ 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 $\dot{V}(t)+\nabla _{U(t)}V(t)-\alpha ^{2}[ \nabla U(t)] ^{t}\cdot \Delta U(t)=-\text{grad } p(t)$ where $$V=(1-\alpha ^{2}\Delta)U$$, $$\text{div} U=0$$.
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 $$\mathcal D_\mu ^s(M)$$ which is expressed as $\langle X,Y\rangle _{1}=\langle X,(1+\text{Ric})Y\rangle _{L^{2}}+\langle \nabla X,\nabla Y\rangle _{L^{2}}$ for $$X,Y\in T_e\mathcal D_\mu ^s(M)$$. The following problems are investigated: $$H^1$$-covariant derivative and its geodesic flow on $$\mathcal D_\mu ^s(M)$$, curvature of the $$H^1$$-metric, existence and uniqueness results for the Jacobi equation, stability and curvature.

##### MSC:
 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 76M30 Variational methods applied to problems in fluid mechanics 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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##### References:
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