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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 $\cal D_\mu ^s(M)$ the group of volume preserving diffeomorphisms of $M$. The weak $L^2$ right invariant Riemannian metric on $\cal D_\mu ^s(M)$ is given by $$ \langle X,Y\rangle _{L^{2}}=\int_{M}\langle X(x),Y(x)\rangle \mu (x),$$ where $X,Y\in T_e\cal D_\mu ^s(M)$ 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 $\cal D_\mu ^s(M)$ are motions of incompressible ideal fluids. {\it D. D. Holm, J. E. Marsden} and {\it 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 $\cal D_\mu ^s(M)$ 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 $X,Y\in T_e\cal D_\mu ^s(M)$. The following problems are investigated: $H^1$-covariant derivative and its geodesic flow on $\cal D_\mu ^s(M)$, 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
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References:
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