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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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