Fredholm properties of Riemannian exponential maps on diffeomorphism groups

Misiołek, Gerard; Preston, Stephen C.

doi:10.1007/s00222-009-0217-3

Cited by 63 publications

(125 citation statements)

References 40 publications

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“…It is then easy to check that in all possible cases, the term (17) can be bounded in terms of the F m norm of ξ . We can similarly prove for m ≥ 1 that η ∈ K m+1 and inf|η | > 0 implies ξ ∈ F m , by using formula (16). We note as a consequence that when ξ is purely imaginary (corresponding to η ∈ A m ), we get a bijective correspondence between an open subset of A m+1 and an open subset of {0} × F m [R] for any m ≥ 1.…”

Section: The Manifold Structure Of the Arc Space A Kmentioning

confidence: 88%

“…The reason for this is that the connection of a submanifold is obtained from the connection on the full manifold by orthogonal projection, and the orthogonal projection operator is not smooth (using the metric (20)). In other situations, when we get a smooth ODE on a submanifold, it is due to smoothness of the orthogonal projection: see, for example, [7] and [16].…”

Section: The Geodesic Equationmentioning

confidence: 97%

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The geometry of whips

Preston

2011

Ann Glob Anal Geom

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In this article, we study geometric aspects of the space of arcs parameterized by unit speed in the L 2 metric. Physically, this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic equation is the nonlinear, nonlocal wave equation η tt = ∂ s (σ η s ), with |η s | ≡ 1 and σ given by σ ss − |η ss | 2 σ = −|η st | 2 , with boundary conditions σ (t, 1) = σ (t, −1) = 0 and η(t, 0) = 0. We prove that the space of arcs is a submanifold of the space of all curves, that the orthogonal projection exists but is not smooth, and as a consequence we get a Riemannian exponential map that is continuous and even differentiable but not C 1 . This is related to the fact that the curvature is positive but unbounded above, so that there are conjugate points at arbitrarily short times along any geodesic.

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Section: The Manifold Structure Of the Arc Space A Kmentioning

confidence: 88%

Section: The Geodesic Equationmentioning

confidence: 97%

The geometry of whips

Preston

2011

Ann Glob Anal Geom

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“…Since K u 0 is compact by Proposition 3.2, we know (d exp e ) tu 0 will be a Fredholm operator, and hence the exponential map will be a non-linear Fredholm map. As a consequence [10] we obtain that conjugate points are of finite multiplicity and form a discrete set along any geodesic, along with various other analogues of theorems in global Riemannian geometry which would otherwise fail in infinite dimensions.…”

Section: The Exponential Mapmentioning

confidence: 93%

“…(See[6,10].) Suppose we have a Lie group G with a right-invariant metric and a smooth geodesic η(t) with η(0) = e and η(0) = u 0 .…”

mentioning

confidence: 99%

Geometry of the contactomorphism group

Chhay¹,

Preston²

2015

Differential Geometry and its Applications

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In this paper we examine the Riemannian geometry of the group of contactomorphisms of a compact contact manifold. We compute the sectional curvature of D θ (M ) in the sections containing the Reeb field and show that it is non-negative. We also solve explicitly the Jacobi equation along the geodesic corresponding to the flow of the Reeb field and determine the conjugate points. Finally, we show that the Riemannian exponential map is a non-linear Fredholm map of index zero.

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“…As is usual when studying diffeomorphism groups (Ebin and Marsden 1970;Misiołek and Preston 2010), the Fréchet manifold structure leads to analytical difficulties when studying geometry due to the lack of an Inverse Function Theorem and to the possibility of non-integrability of vector fields. Hence we enlarge the group to the set of diffeomorphisms η ∈ D s (M) of Sobolev class H s for s > dim M/2 + 1 = n + 3/2, consisting of those maps whose derivatives up to order s are square-integrable in every coordinate chart of compact support.…”

Section: The Riemannian Structure Of the Contactomorphism Groupmentioning

confidence: 99%

Riemannian Geometry of the Contactomorphism Group

Ebin

Preston

2014

Arnold Math J.

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Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the L 2 inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an "Euler-Arnold" equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a "quasi-Lipschitz" estimate on the stream function, which leads to a Beale-Kato-Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler-Arnold equations are the Camassa-Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.

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Fredholm properties of Riemannian exponential maps on diffeomorphism groups

Cited by 63 publications

References 40 publications

The geometry of whips

The geometry of whips

Geometry of the contactomorphism group

Riemannian Geometry of the Contactomorphism Group

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