# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Geometry and curvature of diffeomorphism groups with ${H}^{1}$ metric and mean hydrodynamics. (English) Zbl 0933.58010

Let $\left(M,〈·,·〉\right)$ be a closed oriented Riemannian manifold and ${𝒟}_{\mu }^{s}\left(M\right)$ the group of volume preserving diffeomorphisms of $M$. The weak ${L}^{2}$ right invariant Riemannian metric on ${𝒟}_{\mu }^{s}\left(M\right)$ is given by

${〈X,Y〉}_{{L}^{2}}={\int }_{M}〈X\left(x\right),Y\left(x\right)〉\mu \left(x\right),$

where $X,Y\in {T}_{e}{𝒟}_{\mu }^{s}\left(M\right)$ 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 ${𝒟}_{\mu }^{s}\left(M\right)$ 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

$\stackrel{˙}{V}\left(t\right)+{\nabla }_{U\left(t\right)}V\left(t\right)-{\alpha }^{2}{\left[\nabla U\left(t\right)\right]}^{t}·{\Delta }U\left(t\right)=-\text{grad}\phantom{\rule{4.pt}{0ex}}p\left(t\right)$

where $V=\left(1-{\alpha }^{2}{\Delta }\right)U$, $\text{div}U=0$.

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 ${𝒟}_{\mu }^{s}\left(M\right)$ which is expressed as

${〈X,Y〉}_{1}={〈X,\left(1+\text{Ric}\right)Y〉}_{{L}^{2}}+{〈\nabla X,\nabla Y〉}_{{L}^{2}}$

for $X,Y\in {T}_{e}{𝒟}_{\mu }^{s}\left(M\right)$. The following problems are investigated: ${H}^{1}$-covariant derivative and its geodesic flow on ${𝒟}_{\mu }^{s}\left(M\right)$, 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 (fluid mechanics) 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds