On the group of diffeomorphisms on a compact manifold

Omori, Hideki

doi:10.1090/pspum/015/0271983

Cited by 65 publications

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“…Typically, X is modelled on a Frechét space of smooth sections of a vector bundle over a closed manifold and is a Hilbert Lie group. Inverse limit Hilbert manifolds and inverse limit Hilbert groups, introduced by Omori [20,21], provide an appropriate setting for the study of the YangMills and Seiberg-Witten field equations. (2) Another area of application is the geometric setting of string theory (see Albeverio et al [2], Deligne et al [5], Nag and Sullivan [19]).…”

Section: Areas For Applicationmentioning

confidence: 99%

“…(4) The group D of orientation preserving smooth diffeomorphisms of a compact manifold M is homeomorphic to the product of the group of volume preserving diffeomorphisms D µ , of a volume element µ on M , times the set V of all volumes v > 0 with v = µ. In this case, D µ can be realized as a projective limit of Hilbert-modelled manifolds (see Omori [20,21]) and forms the appropriate framework for the study of hydrodynamics of an incompressible fluid. More precisely, the motion of a perfect incompressible fluid is a geodesic curve η t of D µ with respect to the right invariant metric on D µ , which at e ∈ D µ is (X, Y ) = M X(m), Y (m) m µ(m).…”

Section: Areas For Applicationmentioning

confidence: 99%

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A generalized second-order frame bundle for Fréchet manifolds

Dodson

Galanis

Vassiliou

2005

Journal of Geometry and Physics

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Abstract. Working within the framework of Fréchet modelled infinite dimensional manifolds, we propose a generalized notion of second order frame bundle. We revise in this way the classical notion of bundles of linear frames of order two, the direct definition and study of which is problematic due to intrinsic difficulties of the space models. However, this new structure keeps all the fundamental characteristics of a frame bundle: It is a principal Fréchet bundle associated (differentially and geometrically) with the corresponding second order tangent bundle.

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Section: Areas For Applicationmentioning

confidence: 99%

Section: Areas For Applicationmentioning

confidence: 99%

A generalized second-order frame bundle for Fréchet manifolds

Dodson

Galanis

Vassiliou

2005

Journal of Geometry and Physics

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“…to the operator Φ, because it does not hold in general for Frechet space. For this reason, by utilizing the so-called I.L.H.-method introduced in Λ H. Omori [7] and the regularity of solutions of a non-linear elliptir equation (see A. Douglis-L. Nirenberg [1]), we prove a modified Implicit Function Theorem in certain Frechet spaces with the C°°-topology, which is a unified method of non-linear global analysis and infinite dimensional geometry (see [2], [5]). …”

Section: S (mentioning

confidence: 99%

“…Then the Frechet manifold Γ(E) is called an /.L.H.-manifold (the inverse limit of Hubert manifolds; see [7]). From the local triviality of fiber bundle π: E -> X, it follows that the projection π is a submersion (namely, its differential dπ\ TE-+TX is surjective), whence that Ker(dπ) is a vector subbundle of the tangent bundle TE of E. This vector bundle over E will be denoted by TF(E), and is usually called the tangent bundle along the fiber of E. If s e Γ(E), then the induced bundle s~ιTF(E) by s is a vector bundle over M which we denote simply by T S (E).…”

Section: S (mentioning

confidence: 99%

Non-linear elliptic operators on a compact manifold and an implicit function theorem

Sunada

1975

Nagoya Mathematical Journal

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Many problems in differential and analytic geometry seem to have something to do with the study of non-linear partial differential equations of elliptic type. For instance, the classical Weyl and Minkowski problems for a convex surface have been studied by H. Weyl, H. Lewy, and L. Nirenberg using the iteration method for the construction of solutions of certain non-linear equations of elliptic type (see [6]). Also, M. Kuranishi [3] constructed the effective complete family of deformations of complex analytic structures on a given compact complex manifold as the solution space of another non-linear equation of elliptic type; whereby the basic idea in his work is to apply an implicit function theorem to the non-linear operator of a Banach space, and to construct the bifurcation of solutions explicitly.

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“…The group 3D has the structure of a differentiable manifold modelled on a Fréchet space and with this structure, the group operations are smooth. See Leslie [5] and Omori [8], for the proof in case M has no boundary. Following Omori, we call 3D an ILH Lie group.…”

Section: Introductionmentioning

confidence: 99%

Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid

Ebin¹,

Marsden²

1969

Bull. Amer. Math. Soc.

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1. Introduction. We announce several results on the structure of the group of diffeomorphisms 3) of a compact «-manifold M, possibly with boundary. The group 3D has the structure of a differentiable manifold modelled on a Fréchet space and with this structure, the group operations are smooth. See Leslie [5] and Omori [8], for the proof in case M has no boundary. Following Omori, we call 3D an ILH Lie group.We shall show that several infinite dimensional subgroups of 3D are actually (ILH) submanifolds and hence also have the structure of ILH Lie groups. Also, we construct certain (weak) Riemannian structures on 3D (and on certain subgroups) and find the geodesic flows associated to them.More specifically, if fi is a smooth volume on M and 3D M consists of all diffeomorphisms leaving fi invariant then 3D M is a closed submanifold of 3D. This group 3D,, hence has an ILH Lie group structure. This group is of fundamental importance in the study of hydrodynamics of perfect fluids in ikf, as it is the appropriate configuration space. Arnold [l] has shown that the solution of the classical Euler equations for the fluid corresponds to geodesies on 3D M relative to a given invariant metric on 3D M induced by a Riemannian structure on M. (See Marsden and Abraham [ô] for a precise proof of Arnold's theorem.) We prove the existence of such a smooth geodesic flow and hence that the Euler equations have unique smooth solutions, existing for short time, and varying smoothly with respect to the initial conditions. We expect that the flow is complete and thus the solutions can be uniquely extended for all time. Weaker versions of the above result were proven by Kato [4] and others for two and three dimensional regions in Euclidean space, although Kato [4] shows that in the two dimensional case, solutions do exist for all time.We are indebted to many people for their remarks and suggestions ; especially J. Dowling, T. Kato, I. Kupka, R. Palais and C. Robinson.

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On the group of diffeomorphisms on a compact manifold

Cited by 65 publications

References 0 publications

A generalized second-order frame bundle for Fréchet manifolds

A generalized second-order frame bundle for Fréchet manifolds

Non-linear elliptic operators on a compact manifold and an implicit function theorem

Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid

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