Geodesic Flows on the Diffeomorphism Group of the Circle

(1)

Geodesic Flows on the

Diffeomorphism Group of the Circle

Boris KOLEV

CNRS & Aix-Marseille University

Bergen, Norway, June 2013

Summary. — These notes have been written for theNorwegian Summer School on Analysis and Geometry, which took place at Bergen (Norway) in June 2013. The goal was to provide a basic introduction to the topic of geodesic flows for right- invariant metrics on the diffeomorphism group of the circle Diff^∞(S¹). After an introduction which will permit to familiarise with geodesic flows on Lie groups and the general Euler equation, we will focus on the analytical aspect of the theory in the case of Diff^∞(S¹). We will study in particular smoothness of the metric and the spray defined by a (non-necessary local) inertia operator and the local and global existence of geodesics.

E-mail : [email protected]

Homepage : http://www.cmi.univ-mrs.fr/~kolev/

○c This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

(2)

Introduction

It is a fundamental observation due to Arnold, that the Euler equations of an ideal fluid on a compact Riemannian manifold 𝑀 can be recast as the geodesic flow for a right invariant Riemannian metric on the group of volume preserving diffeomorphisms of 𝑀.

Thereafter, it was discovered that many equations in Mathematical Physics could be obtained the same way. Many of these equations are issued from the classical water-wave problem, where the general governing equations are too complicated to be used in practice by engineers, for instance. Therefore, starting in the XIXth century, many approximations models have been derived to enable a more tractable access of some (possibly partial) relevant physical phenomena.

Several of these models (in particular, the most popular) have been recast asgeodesic equa- tions on the diffeomorphism group of the circle (or related infinite dimensional manifolds like associated homogeneous spaces or central extensions like the Bott-Viraosro group). They allow to use the rich structure of Riemannian geometry to analyse their properties and give a nice interpretation of their behaviour. More recently, interest in this field has emerged in the image processing community, and the subject seems to be very active.

The aim of these lectures is to present a survey of this approach, restricting to Diff^∞(S¹), the diffeomorphism group of the circle (periodic case). We will present, in particular, some geometrical and analytical results on the geodesic flows for some right-invariant metrics on Diff^∞(S¹), especially for somenon-local inertia operators.

The restriction to Diff^∞(S¹) is due only for simplicity of the presentation. All the presented results extend almost directly to the non-periodic case, using instead of Diff^∞(S¹) the regular Lie group Diff_𝐻^∞(R). Furthermore, it seems that important parts of the present work can be extended to higher dimension.

History

The problem under consideration has a long history going back to the eighteenth century, which is worth to recall. I apologized in advance to many authors because the mentioned spot lights will miss many very interesting contributions and are here only to give some landmarks.

∙ 1765: Leonhard Euler reformulated the free motions of a rigid body as the geodesics of a left invariant metric (the kinetic energy) on the Euclidean group (configuration space of a rigid body). The original paper was published in French in Berlin. Euler was very proud of his discovery, that he called in his own words (in French): “la belle propriété des trois axes principaux”, which could be translated by “the beautiful property of the three principal axis”.

∙ 1901: The theory was extended by Henri Poincaré for a general Lagrangian system when the underlying configuration space has the structure of a Lie group, an object which was recently introduced by Sophus Lie. Mentioning Poincaré own words: “Having had the opportunity to study the rotational motion of a hollow rigid body, whose cavity is filled with liquid, I was led to put the general equations of mechanics in a form that I think

(4)

is new and interesting to make know.” (free translation from the original statement in French¹).

∙ 1966: Vladimir Arnold [1] applied this paradigm to the evolution equations of an ideal fluid (with fixed boundary) and recast them as thegeodesic equations for a right-invariant metric on the group of volume-preserving diffeomorphisms of the domain. It could be interesting to remind that his paper was published (in French) for the bi-century of Euler’s paper. His approach was however deliberatelyformal as stated by himself², due probably to the numerous analytical difficulties which arise when one works with infinite dimensional geometry, which were not considered in this pioneering paper, and which could have obscured the main ideas.

∙ 1970: Ebin & Marsden published a long paper [14] where they succeeded to give Arnold’s original ideas, a rigourous foundation and introduced the idea of Hilbert approximation manifolds of the infinite dimensional Fréchet Lie group of smooth diffeomorphisms. Fol- lowing their approach, if we can prove local existence and uniqueness of geodesics (ODE) on diffeomorphism groups then the PDE (Euler equation) is well-posed. Their main con- tribution concerns the local existence of geodesics for the right-invariant𝐿²-metric on the group of smooth volume preserving diffeomorphisms of a compact manifold.

Outline of the lectures

The topics of my four lectures are the following:

1. In the first lecture, I will recall some basic material in differential geometry on Banach manifolds, especially weak and strong metrics, geodesic spray, Lie group actions, momen- tum maps and Euler equations. I will finally introduce (at a more or less formal level) the diffeomorphism group of the circle Diff^∞(S¹) and Euler equations on this group.

2. The goal of this lecture will be to study under which conditions, a right-invariant metric on Diff^∞(S¹) and its spray can be extended smoothly to the Hilbert approximations manifolds 𝒟^𝑞(S¹). We will provide, in particular a criteria on the inertia operator 𝐴 (satisfied by almost all known examples) which ensures both thesmoothness of the metric and the spray on 𝒟^𝑞(S¹).

3. In this lecture, I will discuss local and global existence of geodesics, both in the Hilbert approximations 𝒟^𝑞(S¹) and on the Fréchet Lie group Diff^∞(S¹). I will show in particular that geodesics for the metric𝐻^𝑠 on Diff^∞(S¹) with𝑠 >3/2 are defined for all times.

4. In this last lecture, I will discuss of theRiemannian exponential map of some weak right- invariant metrics on Diff^∞(S¹) and its Hilbert approximations 𝒟^𝑞(S¹). Related to this is the existence of a normal neighbourhood and the Gauss lemma. I will discuss, in particular, the problem of wether the geodesic between two nearby points actually minimizes the energy, and if it is the case, wether it is alocal or a global minimum. Finally, if I have time, I will talk about right-invariant metrics on homogeneous spaces of the group Diff^∞(S¹).

1“Ayant eu l’occasion de m’occuper du mouvement de rotation d’un corps solide creux, dont la cavité est remplie de liquide, j’ai été conduit à mettre les équations générales de la Mécanique sous une forme que je crois nouvelle et qu’il peut être intéressant de faire connaître.” [38]

2“Dans ce qui suit, j’ai tâché, conformément à l’appel de N. Bourbaki, de substituer toujours les calculs aveugles aux idées lucides d’Euler.”

(5)

Chapter 1

Geodesic Flows on a Lie Group and Euler Equations

In this chapter, we recall some basic material in differential geometry on Banach manifolds, especially weak and strong metrics, geodesic spray, Lie group actions, momentum maps and Euler equations. In the last two sections, we introduce the diffeomorphism group of the circle Diff^∞(S¹) together with geodesic flows defined by a right-invariant metric and their corresponding Euler equations.

1.1 Metrics and Geodesic Sprays

For general materials on Banach manifolds, we refer to [31]. Let 𝑀 be a Banach manifold modelled over a Banach space 𝐸.

Definition 1.1. A Riemannian metric 𝑔 on 𝑀 is a smooth, symmetric, positive definite, covariant 2-tensor field on 𝑀.

In other words, we have for each 𝑥 ∈ 𝑀 a symmetric, positive definite, bounded, bilinear form 𝑔(𝑥) on 𝑇_𝑥𝑀 and, in any local chart 𝑈, the mapping

𝑥→𝑔(𝑥), 𝑈 → ℒ²_sym(𝐸,R)

is smooth. Given any 𝑥∈𝑀, we can then consider the continuous, linear operator ℎ_𝑥 :𝑇_𝑥𝑀 →𝑇_𝑥^*𝑀, 𝜉↦→𝑔(𝑥)(𝜉,·)

and called the flat map. The mapping ℎ:𝑇 𝑀 →𝑇^*𝑀 is a vector bundle morphism.

Definition 1.2. The metric is strong if ℎ𝑥 is a topological linear isomorphism for all 𝑥 ∈𝑀, whereas it is weak ifℎ_𝑥 is only injective for all𝑥∈𝑀.

The diffeomorphism group Diff^∞(𝑀) of 𝑀 acts naturally on 𝑇 𝑀. In a local chart 𝑈, the action of 𝜙∈Diff^∞(𝑀) on 𝑇 𝑀 is given by

𝑇 𝜙.(𝑥, 𝑣) = (𝜙(𝑥), 𝜕_𝑥𝜙(𝑥).𝑣), 𝑥∈𝑈, 𝑣 ∈𝐸,

Definition 1.3. An isometry of the Riemannian manifold (𝑀, 𝑔) is a diffeomorphism 𝜙 ∈ Diff^∞(𝑀) such that𝜙^*𝑔=𝑔, where

(𝜙^*𝑔)(𝜉1, 𝜉2) :=𝑔(𝜙(𝑥))(𝑇 𝜙.𝜉1, 𝑇 𝜙.𝜉2), 𝜉1, 𝜉2∈𝑇𝑥𝑀, 𝑥∈𝑀.

(6)

On the cotangent bundle 𝑇^*𝑀 of 𝑀, there exists a canonical 1-form Θ called theLiouville form and defined by

Θ_𝛼(𝑋) :=𝛼(𝑇 𝜋(𝑋)), 𝛼∈𝑇^*𝑀, 𝑋 ∈𝑇_𝛼(𝑇^*𝑀),

where 𝑇 𝜋 : 𝑇(𝑇^*𝑀) → 𝑇 𝑀 is the tangent map of the canonical projection 𝜋 : 𝑇^*𝑀 → 𝑀. Given a local chart𝑈 in𝑀, Θ is simply given by the formula

Θ_𝛼(𝑋) =𝜆(𝑢),

where 𝛼 = (𝑥, 𝜆), 𝑋 = (𝑥, 𝜆, 𝑢, 𝜔), 𝑥 ∈ 𝑈, 𝑢 ∈ 𝐸, and 𝜆, 𝜔 ∈ 𝐸^*. The exterior derivative Ω := 𝑑Θ of Θ defines a symplectic¹ form on 𝑇^*𝑀: the canonical symplectic structure on the cotangent bundle. Given a local chart𝑈 in𝑀, Ω is given by the formula

Ω_𝛼(𝑋₁, 𝑋₂) =𝜔₁(𝑢₂)−𝜔₂(𝑢₁),

where 𝛼 = (𝑥, 𝜆), 𝑋_𝑖 = (𝑥, 𝜆, 𝑢_𝑖, 𝜔_𝑖), 𝑥∈𝑈, 𝑢_𝑖 ∈𝐸, and 𝜆, 𝜔_𝑖 ∈𝐸^*. The canonical symplectic structure on 𝑇^*𝑀 is strong if the model Banach space 𝐸 isreflexive.

The pull-back of the Liouville form Θ by the flat mapℎ:𝑇 𝑀 →𝑇^*𝑀 defines a 1-form Θ^𝑔 on 𝑇 𝑀 which is given by

Θ^𝑔_𝜉(𝑋) :=𝑔(𝑥)(𝜉, 𝑇 𝜋(𝑋)), 𝜉∈𝑇 𝑀, 𝑋 ∈𝑇_𝜉(𝑇 𝑀).

Given a local chart𝑈 in𝑀, Θ^𝑔 is given by the formula Θ^𝑔_𝜉(𝑋) =𝑔(𝑥)(𝑣, 𝑢), where 𝜉= (𝑥, 𝑣),𝑋 = (𝑥, 𝑣, 𝑢, 𝑤), 𝑥∈𝑈 and𝑢, 𝑣, 𝑤 ∈𝐸.

Remark 1.4. The Liouville form Θ on𝑇^*𝑀 is invariant by all diffeomorphisms² but Θ^𝑔 is only invariant by isometriesof 𝑔.

The exterior derivative Ω^𝑔 := 𝑑Θ^𝑔 is asymplectic form on 𝑇 𝑀. It is strong if the metric is strong and weak if the metric is weak. Given a local chart 𝑈 in 𝑀, Ω^𝑔 is given by the formula

Ω^𝑔_𝜉(𝑋1, 𝑋2) =𝜕𝑥𝑔(𝑥).𝑢1(𝑣, 𝑢2) +𝑔(𝑥)(𝑤1, 𝑢2)−𝜕𝑥𝑔(𝑥).𝑢2(𝑣, 𝑢1)−𝑔(𝑥)(𝑤2, 𝑢1), where 𝜉= (𝑥, 𝑣),𝑋𝑖= (𝑥, 𝑣, 𝑢𝑖, 𝑤𝑖),𝑥∈𝑈 and 𝑣, 𝑢𝑖, 𝑤𝑖 ∈𝐸.

∙ When the metric is strong, we can therefore associate to each function 𝐻 on 𝑇 𝑀 a sym- plectic gradient orHamiltonian vector field 𝐹_𝐻 on 𝑇 𝑀, defined by

𝑑_𝜉𝐻.𝑋 :=−Ω^𝑔_𝜉(𝐹𝐻(𝜉), 𝑋), (1.1)

for all 𝑋∈𝑇𝜉(𝑇 𝑀) and all𝜉 ∈𝑇 𝑀.

∙ When the metric is weak, given a function 𝐻 on 𝑇 𝑀, the Hamiltonian vector field may not exist but if it exists, it is uniquely defined by (1.1).

1A symplectic form on a Banach manifold is a closed 2-form which is non-degenerate. It isstrongif it induces an isomorphism with the tangent bundle while it isweak if it is only injective.

2The action of𝜙∈Diff^∞(𝑀) on𝑇^*𝑀 is defined locally by

𝑇^*𝜙.(𝑥, 𝜆) = (𝜙(𝑥), 𝜆(𝜕𝑥𝜙⁻¹(𝜙(𝑥)).𝑣)), 𝑥∈𝑈, 𝜆∈𝐸^*.

(7)

Definition 1.5. Thegeodesic spray 𝐹 of the metric𝑔is defined (if it exists) as the Hamiltonian vector field of the energy function

𝐾(𝜉) := 1

2𝑔(𝑥)(𝜉, 𝜉).

The integral curves of𝐹 are called thegeodesics.

Remark 1.6. The spray𝐹is asecond order vector field, which means that not only (𝜋_{𝑇 𝑀}∘𝐹)(𝜉) = 𝜉, but also (𝑇 𝜋𝑀∘𝐹)(𝜉) =𝜉. In particular, given an integral curve𝜉(𝑡) of𝐹, we get

𝑑

𝑑𝑡𝜋_𝑀(𝜉) =𝜉.

We will therefore refer indifferently to𝜉(𝑡) or to its base-path𝑥(𝑡) :=𝜋𝑀(𝜉(𝑡)) as a geodesic.

In a local chart𝑈×𝐸 of 𝑇 𝑀, the Hamiltonian vector field 𝐹 is given by 𝐹(𝑥, 𝑣) := (𝑥, 𝑣, 𝑣, 𝑆(𝑥, 𝑣)),

where 𝑥∈𝑈,𝑣∈𝐸 and 𝑆(𝑥, 𝑣) is defined implicitly by 𝑔(𝑥)(𝑆(𝑥, 𝑣), 𝑢) = 1

2𝜕𝑥𝑔(𝑥).𝑢(𝑣, 𝑣)−𝜕𝑥𝑔(𝑥).𝑣(𝑣, 𝑢), ∀𝑢∈𝐸. (1.2) Remark 1.7. Although, we will not make so much use of covariant derivatives in these lectures, it is worth to note that the existence of the geodesic spray for a weak Riemannian metric implies the existence of a unique covariant derivative∇compatible with the metric (see [31]). In a local chart, this covariant is defined as

∇_𝜉𝑋 :=𝜕𝑢(𝑥).𝑣−𝐵(𝑢(𝑥), 𝑣) where 𝜉= (𝑥, 𝑣) and𝑋(𝑥) = (𝑥, 𝑢(𝑥)) and𝐵(𝑢, 𝑣) :=𝜕_𝑣²𝑆(𝑥,0).

Exercise 1.1. Show that the geodesics are indeed the extremal curves of the energy functional 𝐸(𝛾) :=

ˆ _𝑏

𝑎

𝐾( ˙𝛾)𝑑𝑡.

Proposition 1.8. The geodesic spray (if it exists) is invariant under any isometry of the metric 𝛾. More precisely

𝐹(𝑇 𝜙.𝜉) =𝑇(𝑇 𝜙).𝐹(𝜉) for any diffeomorphism 𝜙such that 𝜙^*(𝑔) =𝑔.

Proof. This is a consequence of the defining formula (1.1) for the spray and the fact that both the energy function 𝐾 and the symplectic form Ω^𝑔 are invariant under an isometry.

1.2 Lie Groups

Definition 1.9. ALie groupis a set𝐺equipped with a group structure together with a smooth manifold structure and such that the group operations

(𝑔, ℎ)↦→𝑔ℎ, and 𝑔↦→𝑔⁻¹ are smooth.

(8)

Canonical actions. On a Lie group𝐺, there are three natural actions of 𝐺on itself:

1. The right action defined by

𝑅𝑔 :𝐺→𝐺, ℎ↦→ℎ𝑔, 𝑔, ℎ∈𝐺, 2. The left action defined by

𝐿𝑔 :𝐺→𝐺, ℎ↦→𝑔ℎ, 𝑔, ℎ∈𝐺, 3. The inner action defined by

𝐼_𝑔:𝐺→𝐺, ℎ↦→𝑔ℎ𝑔⁻¹, 𝑔, ℎ∈𝐺.

Remark 1.10. Notice that the right action and the left actions commutes and that the inner action consists ofautomorphisms of 𝐺:

𝐼𝑔(ℎ1ℎ2) =𝐼𝑔(ℎ1)𝐼𝑔(ℎ2), 𝑔, ℎ1, ℎ2 ∈𝐺.

In particular, 𝐼𝑔(𝑒) =𝑒.

Lie group algebra. The tangent space to a Lie group 𝐺 at the unit element𝑒, denoted by g plays a special role. It is equipped with a Lie algebra structure³, that is a skew symmetric bracket

g×g→g, (𝑢, 𝑣)↦→[𝑢, 𝑣]

which satisfiesJacobi identity

[𝑢,[𝑣, 𝑤]] + [𝑣,[𝑤, 𝑢]] + [𝑤,[𝑢, 𝑣]] = 0, ∀𝑢, 𝑣, 𝑤∈g.

The space gis called theLie algebra of the group𝐺.

Remark 1.11. A Lie group 𝐺 is equipped with a canonical vector-valued 1-form, called the Maurer-Cartan form and defined by

𝜔𝑔(𝜉𝑔) =𝑇 𝐿_𝑔⁻¹𝜉𝑔, 𝜉𝑔 ∈𝑇𝑔𝐺 which shows that the tangent bundle to 𝐺is trivial𝑇 𝐺≃𝐺×g.

The Lie group exponential. To each element𝑢of the Lie group algebrag, we can associate the first order ODE

𝑔𝑡=𝑇 𝑅𝑔𝑢, 𝑔(0) =𝑒. (1.3)

Exercise 1.2. Show that the solution of this ODE is defined for all time and that we get the same curve if we replace 𝑇 𝑅𝑔𝑢 by 𝑇 𝐿𝑔𝑢 in the equation.

Definition 1.12. The solution of equation (1.3) will be denoted by exp(𝑡𝑢) and𝑢↦→exp(𝑢) is called the group exponential.

Adjoint action of 𝐺. The tangent map at𝑒of the inner automorphism𝐼_𝑔 is denoted by Ad_𝑔 and defines alinear representation of the group𝐺in its Lie algebra g, i.e.

Ad_𝑔ℎ= Ad_𝑔Ad_ℎ

and called the adjoint representation of 𝐺. It preserves the Lie bracket [Ad_𝑔𝑢,Ad_𝑔𝑣] = Ad𝑔[𝑢, 𝑣], 𝑢, 𝑣∈g, 𝑔∈𝐺.

3A left-invariant vector field is uniquely defined by its value at the unit element𝑒. Since the Lie bracket of such fields is again a left-invariant vector field, the Lie algebra structure on vector fields induces a Lie algebra structure ong.

(9)

Adjoint action of g. This representation of 𝐺ong induces a (Lie algebra) representation of gong, called theadjoint representationofg. It is defined as the differential of the map𝑔↦→Ad_𝑔 at𝑒, and denoted by ad_𝑢.

Exercise 1.3. Prove that ad_𝑢𝑣= [𝑢, 𝑣] for all 𝑢, 𝑣∈g.

Coadjoint action of 𝐺. Let g^* be the dual vector space to the Lie algebra g (elements ofg^* are linear functionals on g). The coadjoint representation of 𝐺on g^* is defined by

Ad^*_𝑔 :g^* →g^*, 𝑚↦→𝑚∘Ad_𝑔−1, for every 𝑔∈𝐺and 𝑚∈g^*.

Remark 1.13. The choice of𝑔⁻¹ in the definition of Ad^*_𝑔 ensures that𝐴𝑑^*is aleft representation, that is

Ad^*_𝑔ℎ= Ad^*_𝑔Ad^*_ℎ rather than the opposite (right representation).

Coadjoint action of g. Similarly, we define the coadjoint representation of g on g^* as the infinitesimal version of Ad^*. More precisely

ad^*_𝑢:g^*→g^*, 𝑚↦→ −𝑚∘ad_𝑢, where 𝑢∈g and𝑚∈g^*.

Isometric actions of a Lie group 𝐺on a Riemannian manifold (𝑀, 𝑔). Consider now a smooth isometric action of a Lie group𝐺on a Riemannian manifold (𝑀, 𝑔). By this, we mean a smooth map

𝜓:𝐺×𝑀 →𝑀, (ℎ, 𝑥)↦→ℎ·𝑥 such that

1. 𝜓(ℎ,·) :𝑀 →𝑀 is an isometry of the metric 𝑔 on 𝑀 for each ℎ∈𝐺, and 2. the map ℎ↦→𝜓(ℎ,·) from 𝐺to Diff^∞(𝑀) is a group morphism.

Associated to this action of 𝐺 on 𝑀, there is an infinitesimal action of g on 𝑀, which is defined as follows. Given 𝑢∈g, letℎ(𝑠) be a smooth curve in𝐺, such that

ℎ(0) =𝑒, ℎ(0) =˙ 𝑢.

We define a smooth vector field 𝜉_𝑢 on𝑀 by the following formula 𝜉𝑢(𝑥) = 𝑑

𝑑𝑠

⃒

⃒_𝑠=0

𝜓(ℎ(𝑠), 𝑥).

Exercise 1.4. Show that

𝜉_[𝑢,𝑣]= [𝜉_𝑢, 𝜉_𝑣]

(10)

The moment map. Let (𝑀, 𝑔) be a Riemannian manifold and𝐺 be a Lie group which acts isometrically on 𝑀.

Definition 1.14. The moment map is defined as the mapping 𝜇:𝑇 𝑀 →g^*, such that

𝜇(𝜉_𝑥).𝑤:= Θ^𝑔_𝜉(𝑋_𝑤(𝜉_𝑥)) =< 𝜉_𝑥, 𝜉_𝑤(𝑥)>_𝑥, (1.4) where 𝑤∈g,𝑋𝑤 is the infinitesimal action of g on𝑇 𝑀, and𝜉𝑤 is the infinitesimal action ofg on 𝑀.

Exercise 1.5. Show that

𝜇(ℎ·𝜉) = Ad^*_ℎ𝜇(𝜉), 𝜉∈𝑇 𝑀, ℎ∈𝐺, for a left action, and that

𝜇(ℎ·𝜉) = Ad^*_ℎ−1𝜇(𝜉), 𝜉∈𝑇 𝑀, ℎ∈𝐺, for a right action.

Proposition 1.15. Let 𝑥(𝑡) be a geodesic curve. Then 𝜇( ˙𝑥(𝑡))is constant.

Proof. Using Cartan formula (see [31, Chapter 5])

𝐿𝑋𝜃=𝑑𝑖𝑋𝜃+𝑖𝑋𝑑𝜃, we get

𝑑

𝑑𝑡𝜇( ˙𝑥(𝑡)).𝑢=𝑑(𝑖_𝑋_𝑢Θ^𝑔) (¨𝑥) =−Ω^𝑔(𝑋𝑢( ˙𝑥), 𝐹( ˙𝑥)) because Θ^𝑔 is invariant under 𝐺and hence𝐿_𝑋_𝑢Θ^𝑔 = 0 for every𝑢∈g. But

Ω^𝑔(𝑋_𝑢( ˙𝑥), 𝐹( ˙𝑥)) =𝑑_𝑥_˙𝐾.𝑋_𝑢( ˙𝑥) = 0, because 𝐾 is invariant under 𝐺.

Right invariant metric on a Lie group. A right-invariant Riemannian metric on a Lie group 𝐺 is a Riemannian metric which is invariant by all right-translations 𝑅ℎ. Because this action is transitive, such a metric is uniquely defined by its value at the unit element, that is by an inner product on the Lie algebra g of the group. the corresponding flat map at the unit element is an operator

𝐴:g→g^*

which, for historical reasons, going back to the work of Euler [20] on the motion of the rigid body, is called the inertia operator. The value of the flat map 𝐴ℎ at any point ℎ ∈ 𝐺 is then obtained by the formula

𝐴_ℎ =𝑇 𝑅^*_ℎ∘𝐴∘𝑇 𝑅_ℎ⁻¹, where the operator 𝑇 𝑅^*_ℎ :g^* →𝑇_ℎ^*𝐺 is defined by

𝑇 𝑅^*_ℎ𝑚:=𝑚∘𝑇 𝑅_ℎ⁻¹, as illustrated in the following diagram

𝑇_ℎ𝐺 ^𝐴^ℎ ^//

𝑇 𝑅_ℎ−1

𝑇_ℎ^*𝐺

𝑇 𝑅^*_ℎ

g 𝐴 //g^*

(11)

The Euler-Poincaré equation. Given a smooth pathℎ(𝑡) in𝐺, we define its (right)Eulerian velocity (also called the logarithmic derivative), which lies in the Lie algebra g, by

𝑢(𝑡) =𝑇 𝑅_ℎ⁻¹_(𝑡)ℎ(𝑡).˙

The infinitesimal action ofg corresponding to theright action of 𝐺on itself is given by 𝜉_𝑤(ℎ) = 𝑑

𝑑𝑠

⃒

⃒_𝑠=0

ℎexp(𝑠𝑤) =𝑇 𝐿_ℎ.𝑤.

For the moment map (1.14), we obtained therefore the following expression 𝜇(𝜉(𝑡)) =𝜇(ℎ·𝑢(𝑡)) = Ad^*_ℎ−1𝜇(𝑢(𝑡)) = Ad^*_ℎ−1𝑚(𝑡),

where 𝑚(𝑡) = 𝐴𝑢(𝑡). The conservation of the momentum along a geodesic leads then to the following theorem.

Theorem 1.16. If ℎ(𝑡) is a geodesic of a right-invariant metric on 𝐺, then 𝑑

𝑑𝑡Ad^*_ℎ−1(𝑡)𝑚(𝑡) = 0.

Remark 1.17. The preceding theorem is a generalization of the conservation of the angular momentum for a free rigid body.

Corollary 1.18 (Euler-Poincaré equation). If ℎ(𝑡) is a geodesic of a right-invariant metric on 𝐺, then

𝑚𝑡= ad^*_𝑢𝑚 (1.5)

Proof. We have

𝑑

𝑑𝑡Ad^*_ℎ(𝑡)𝑚(𝑡) = (︂𝑑

𝑑𝑡Ad^*_ℎ(𝑡) )︂

𝑚(𝑡) + Ad^*_ℎ(𝑡)𝑚_𝑡(𝑡),

but 𝑑

𝑑𝑡Ad^*_ℎ(𝑡) =−Ad^*_ℎ(𝑡)∘ad^*_𝑢(𝑡), and therefore, by theorem 1.16, we get

𝑚𝑡(𝑡)−ad^*_𝑢(𝑡)𝑚(𝑡) = 0.

Exercise 1.6. Prove that for a left invariant metric, we have 𝑑

𝑑𝑡Ad^*_ℎ(𝑡)𝑚(𝑡) = 0, and 𝑚𝑡(𝑡) =−ad^*_𝑢(𝑡)𝑚(𝑡), where 𝑢(𝑡) =𝑇 𝐿_ℎ⁻¹_(𝑡)ℎ(𝑡).˙

The Euler equation. When the inertia operator𝐴:g→g^* is invertible, the Euler-Poincaré equation (1.5) can be given a contravariant formulation, known as the Euler equation. It gen- eralizes, for any Lie groups, the Euler equation on the angular velocity, derived by Euler in 1765 [20]. Indeed, from equation (1.5), we get for any 𝑣∈g

< 𝑢𝑡, 𝑣 >=𝑚𝑡(𝑣)

= (ad^*_𝑢𝑚)(𝑣)

=−𝑚(ad_𝑢𝑣)

=−< 𝑢,ad_𝑢𝑣 >

(12)

where ad^⊤_𝑢 is theadjoint (relatively to the inner product given by A) of ad_𝑢. We have therefore

𝑢_𝑡=−ad^⊤_𝑢 𝑢, (1.6)

which is called the Euler equation on (right) Eulerian velocity.

Exercise 1.7. Prove that for a left invariant metric, we get 𝑢𝑡= ad^⊤_𝑢 𝑢, where 𝑢(𝑡) =𝑇 𝐿_ℎ⁻¹_(𝑡)ℎ(𝑡) is the left Eulerian velocity.˙

Remark 1.19. Notice that, in order to formulate the Euler equation, the hypothesis that the inertia operator is invertible is too strong. Indeed, the only thing required is that ad^⊤_𝑢 is well- defined or even the weaker hypothesis that the symmetric part of the bilinear operator (𝑢, 𝑣)↦→

ad^⊤_𝑢 𝑣, denoted by 𝐵(𝑢, 𝑣) is well-defined. This operator𝐵 is defined implicitly by

< 𝐵(𝑢, 𝑣), 𝑤 >= 1

2{< 𝑣,ad_𝑢𝑤 >+< 𝑢,ad_𝑣𝑤 >}, for all 𝑢, 𝑣, 𝑤∈g.

Exercise 1.8. Given a Banach Lie group, equipped with a weak right-invariant metric, show that the existence of the bilinear operator 𝐵 on g ensures the existence of the geodesic spray. [Hint:

Show that if 𝜓:𝐺×𝐺→𝐺represents the group composition, then 𝑆(𝑒, 𝑣) :=𝜕₁𝜕₂𝜓(𝑒, 𝑒).𝑣.𝑣−ad^⊤_𝑣 𝑣, 𝑣∈g is the solution of equation (1.2) at 𝑥=𝑒.]

1.3 The Diffeomorphisms Group of the Circle

In this section, we will apply the theory exposed in the preceding section to the case of the group of smooth diffeomorphisms of the circle. Although this group is not a Banach Lie group, it can be endowed with the structure of a Fréchet Lie group [27, 35], modelled on the Fréchet vector space⁴ C^∞(S¹). The preceding results apply if we replace Fréchet differentials with Gâteaux derivatives(or directional derivatives).

The Fréchet Lie group Diff^∞(S¹). Let Diff^∞(S¹) be the group of all smooth and orientation preserving diffeomorphisms on the circle. This group is naturally equipped with a Fréchet manifold structure; it can be covered by charts taking values in theFréchet vector spaceC^∞(S¹) and in such a way that the change of charts are smooth maps (a smooth atlas with only two charts may be constructed, see [26]).

Since the composition and the inverse map are smooth for this structure we say that Diff^∞(S¹) is a Fréchet-Lie group, cf. [27]. Its Lie algebra, Vect(S¹), is the space of smooth vector fields on the circle. It is isomorphic to C^∞(S¹) with the Lie bracket given by

[𝑢, 𝑣] =𝑢_𝑥𝑣−𝑢𝑣_𝑥.

Remark 1.20. Like any Lie group, Diff^∞(S¹) is aparallelizable manifold 𝑇Diff^∞(S¹)∼Diff^∞(S¹)×C^∞(S¹).

4A topological vector space 𝐸 has a canonicaluniform structure. When this structure iscomplete and when the topology of𝐸may be given by a countable family ofsemi-norms, we say that𝐸is a Fréchet vector space. The typical example of a Fréchet space is the space of smooth functions on a compact manifold, where the semi-norms are just the𝐶^𝑘-norms (𝑘= 0,1, . . .). A Fréchet vector space is a Banach space if and only if it is locally bounded, which is not the case of C^∞(S¹).

(13)

Diffeomorphisms of Sobolev Class 𝐻^𝑞. The Fréchet manifold Diff^∞(S¹) has a richest structure. It can be approximated by extendedHilbert manifolds. More precisely, we define the Sobolev space 𝐻^𝑞(S¹) as the completion of C^∞(S¹) for the norm

‖𝑢‖_𝐻𝑞(S¹):=

⎛

⎝

∑︁

𝑛∈Z

(1 +𝑛²)^𝑞|^𝑢_𝑛|²

⎞

⎠

1/2

,

where 𝑞 ∈ R, 𝑞 ≥ 0. We recall that 𝐻^𝑞(S¹) is a multiplicative algebra for 𝑞 > 1/2 (cf. [40, Theorem 2.8.3]). This means that

‖𝑢𝑣‖_𝐻𝑞(S¹) .‖𝑢‖_𝐻𝑞(S¹)‖𝑣‖_𝐻𝑞(S¹), 𝑢, 𝑣∈𝐻^𝑞(S¹).

Definition 1.21. We say that a𝐶¹ diffeomorphism𝜙of S¹ is of class 𝐻^𝑞 if for any of its lifts toR, ˜𝜙, we have

𝜙˜−id∈𝐻^𝑞(S¹).

Proposition 1.22 (Ebin-Marsden, 1970). For 𝑞 >3/2, the set 𝒟^𝑞(S¹) of 𝐶¹-diffeomorphisms of the circle which are of class𝐻^𝑞 has the structure of a Hilbert manifold (modelled on𝐻^𝑞(S¹)) and is a topological group.

More precisely, it was established in [11] (see also [12,13,14]), that

(𝑢, 𝜙)↦→𝑢∘𝜙, 𝐻^𝑞(S¹)× 𝒟^𝑞(S¹)→𝐻^𝑞(S¹) (1.7) is continuous and that

(𝑢, 𝜙)↦→𝑢∘𝜙, 𝐻^𝑞+𝑘(S¹)× 𝒟^𝑞(S¹)→𝐻^𝑞(S¹) (1.8) is of class 𝐶^𝑘. Furthermore, given𝜙∈ 𝒟^𝑞(S¹),

𝑢↦→𝑅𝜙(𝑢) :=𝑢∘𝜙, 𝐻^𝑞(S¹)→𝐻^𝑞(S¹) (1.9) is asmooth map.

Remark 1.23. Notice that 𝒟^𝑞(S¹) is a topological group but not a Lie group; i.e. composition and inversion in 𝒟^𝑞(S¹) are continuous but not differentiable.

The Fréchet Lie group group Diff^∞(S¹) may be viewed as an inverse limit ofHilbert manifolds;

an ILH (inverse limit Hilbert) Lie group. More precisely, we have Diff^∞(S¹) = ^⋂︁

𝑞>³₂

𝒟^𝑞(S¹),

and we call the scales of manifolds𝒟^𝑞(S¹))_𝑞>3/2, a Hilbert manifold approximation of Diff^∞(S¹).

Remark 1.24. Notice that the tangent bundle of the Hilbert manifold𝒟^𝑞(S¹) is trivial. Indeed, let t:𝑇S¹ →S¹×Rbe a trivialisation of the tangent bundle of the circle. Then

Ψ :𝑇𝒟^𝑞(S¹)→ 𝒟^𝑞(S¹)×𝐻^𝑞(S¹), 𝜉 ↦→t∘𝜉

defines a smooth vector bundle isomorphism becauset is smooth (see [14, Page 107]).

(14)

1.4 Euler Equations on Diff

^∞

( S

¹

)

Aright-invariantmetric on Diff^∞(S¹) is defined by an inner product on the Lie algebra Vect(S¹) = C^∞(S¹). In the following, we assume that this inner product is given by

⟨𝑢, 𝑣⟩= ˆ

S¹

(𝐴𝑢)𝑣 𝑑𝑥,

where 𝐴 : C^∞(S¹) →C^∞(S¹) is a 𝐿²-symmetric, positive definite, invertible Fourier multiplier (i.e. a continuous linear operator on C^∞(S¹) which commutes with 𝐷:=𝑑/𝑑𝑥).

By translating the above inner product, we obtain an inner product on each tangent space 𝑇𝜙Diff^∞(S¹)

⟨𝜂, 𝜉⟩_𝜙 =⟨𝜂∘𝜙⁻¹, 𝜉∘𝜙⁻¹⟩_𝑖𝑑= ˆ

S¹

𝜂(𝐴_𝜙𝜉)𝜙_𝑥𝑑𝑥, (1.10)

where 𝜂, 𝜉 ∈ 𝑇𝜙Diff^∞(S¹) and 𝐴𝜙 = 𝑅𝜙 ∘𝐴∘𝑅_𝜙⁻¹, and 𝑅𝜙(𝑣) := 𝑣∘𝜙. This defines a weak Riemannian metric on Diff^∞(S¹) because composition and inversion are smooth on the Fréchet Lie group Diff^∞(S¹). The correspondingflat map is given by

𝑇_𝜙Diff^∞(S¹)→𝑇_𝜙^*Diff^∞(S¹), 𝑣↦→𝜙_𝑥𝐴_𝜙. Lemma 1.25. The mapad^⊤_𝑢 is well defined and given by

ad^⊤_𝑢 𝑤=𝐴⁻¹[2(𝐴𝑤)𝑢_𝑥+ (𝐴𝑤)_𝑥𝑢]

for 𝑢, 𝑤 ∈C^∞(S¹).

Proof. We have

<ad_𝑢𝑣, 𝑤 >=

ˆ

S¹

(𝐴𝑤)(𝑢_𝑥𝑣−𝑢𝑣_𝑥)𝑑𝑥= ˆ

S¹

[2(𝐴𝑤)𝑢_𝑥+ (𝐴𝑤)_𝑥𝑢)]𝑣 𝑑𝑥

where 𝑢, 𝑣, 𝑤∈C^∞(S¹). But since 𝐴: C^∞(S¹)→C^∞(S¹) is supposed to be invertible, we get ad^⊤_𝑢 𝑤=𝐴⁻¹[2(𝐴𝑤)𝑢_𝑥+ (𝐴𝑤)_𝑥𝑢],

which finished the proof.

The corresponding Euler equation on Diff^∞(S¹) is given by

𝑢𝑡=−𝐴⁻¹{(𝐴𝑢)_𝑥𝑢+ 2(𝐴𝑢)𝑢𝑥}. (1.11) An example is furnished by the𝐻^𝑘 inner product on C^∞(S¹) (𝑘≥0) ,

a_𝑘(𝑢, 𝑣) = ˆ

S¹

(︁𝑢𝑣+𝑢𝑥𝑣𝑥+· · ·+𝑢^(𝑘)_𝑥 𝑣^(𝑘)_𝑥 ^)︁𝑑𝑥 𝑢, 𝑣∈C^∞(S¹).

In this case, the inertia operator is just 𝐴_𝑘= 1− 𝑑²

𝑑𝑥² +· · ·+ (−1)^𝑘 𝑑^2𝑘 𝑑𝑥^2𝑘.

Example 1.26. For𝑘= 0 (that is, for the𝐿²-metric), the corresponding Euler’s equation (1.11) is the inviscid Burgers equation

𝑢𝑡+ 3𝑢𝑢𝑥= 0. (1.12)

(15)

Example 1.27. For𝑘= 1 (that is for the𝐻¹-metric), the corresponding Euler’s equation (1.11) is the Camassa-Holm equation

𝑢_𝑡−𝑢_𝑡𝑥𝑥+ 3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥= 0, (1.13) a model in the theory of shallow water waves [6,21].

Corollary 1.28. The weak Riemannian metric onDiff^∞(S¹)induced by𝐴 admits the following geodesic spray

𝐹 : (𝜙, 𝑣)↦→(𝜙, 𝑣, 𝑣, 𝑆𝜙(𝑣)) where

𝑆_𝜙(𝑣) :=^(︁𝑅_𝜙∘𝑆∘𝑅_𝜙⁻¹^)︁(𝑣), and

𝑆(𝑢) :=𝐴⁻¹{[𝐴, 𝑢]𝑢_𝑥−2(𝐴𝑢)𝑢_𝑥}.

Proof. Let𝜙be the flow of the time dependent vector field𝑢 and let 𝑣=𝜙𝑡. Then 𝑣𝑡= (𝑢𝑡+𝑢𝑢𝑥)∘𝜙

and 𝑢 solves the Euler equation (1.11) if and only if (𝜙, 𝑣) is a solution of {︃𝜙𝑡=𝑣,

𝑣𝑡=𝑆𝜙(𝑣), (1.14)

One can check that𝐹 is the corresponding geodesic spray (see exercise1.8).

Suppose now that 𝐴 is a differential operator of order 𝑟. Then, the right hand side of the Euler equation is of order 1 because if 𝑢 ∈ 𝐶^𝑘(S¹) then 𝐴⁻¹[𝑢(𝐴𝑢𝑥)] ∈ 𝐶𝑘−1. Hence the Euler equation cannot be realized as an ODE on the Banach space 𝐶^𝑘(S¹). It is however quite surprising that in Lagrangian coordinates we obtain an ODE, provided that the order of 𝐴 is not less than 1.

The main observation is that if𝐴 is adifferential operator of order𝑟 ≥1 then the quadratic operator

𝑆(𝑢) :=𝐴⁻¹{[𝐴, 𝑢]𝑢_𝑥−2(𝐴𝑢)𝑢𝑥}

is of order 0 because the commutator [𝐴, 𝑢] is of order less than ≤ 𝑟−1. One might expect, that for a larger class of operators 𝐴, the quadratic operator 𝑆 to be of order 0 and the second order system (1.14) to be the local expression of an ODE on the Banach manifold𝑇𝒟^𝑞(S¹). The special case where 𝐴 is a differential operator with constant coefficients has been extensively studied in [7,8,15].

It is the aim of theses lectures to extend these results for a general Fourier multiplier, under certain conditions on its symbol. We will provide, in particular, local existence and uniqueness of the initial value problem for the geodesics of the right-invariant 𝐻^𝑠 metric on the Fréchet- Lie group Diff^∞(S¹), and more generally for a larger class of right-invariant weak Riemannian metric.

(16)

Chapter 2

Smoothness of the Metric and the Spray

The goal of this chapter is to study under which conditions, a right-invariant metric on Diff^∞(S¹) and its spray can be extended smoothly to the Hilbert approximations manifolds 𝒟^𝑞(S¹). We provide, in particular a criteria on the inertia operator 𝐴 (satisfied by almost all known examples) which ensures both the smoothness of the metric and the spray.

2.1 Fourier Multipliers

We define

e_𝑛(𝑥) = exp(2𝜋𝑖𝑛𝑥), for𝑛∈Z and𝑥∈S¹.

Definition 2.1. A Fourier multiplier is a continuous linear operator 𝑃 on the Fréchet space C^∞(S¹,C), which satisfies one of the following three equivalent conditions:

1. 𝑃 commutes with all rotations 𝑅𝑠. 2. [𝑃, 𝐷] = 0, where 𝐷=𝑑/𝑑𝑥.

3. For each𝑛∈Z, there is a𝑝(𝑛)∈Csuch that𝑃e_𝑛=𝑝(𝑛)e_𝑛.

Since every smooth function on the unit circle S¹ can be represented by its Fourier series, we get that

(𝑃 𝑢)(𝑥) =^∑︁

𝑘∈Z

𝑝(𝑘)^𝑢(𝑘)e𝑘(𝑥), (2.1)

for every Fourier multiplier𝑃 and every𝑢∈C^∞(S¹), where 𝑢(𝑘) :=^

ˆ

S¹

𝑢(𝑥)e−𝑘(𝑥)𝑑𝑥,

stands for the 𝑘-th Fourier coefficients of 𝑢. The sequence 𝑝:Z→ Cis called thesymbol of𝑃. We use also the notation 𝑃 :=op(𝑝(𝑘)) for the Fourier multiplier induced by the sequence 𝑝.

Remark 2.2. The space of Fourier multipliers is acommutative subalgebraof the algebra of linear operators on C^∞(S¹,C). It contains all linear differential operators with constant coefficients.

Notice that a Fourier multiplier𝑃 is𝐿²-symmetric iff its symbol 𝑝 is real.

(17)

A Fourier multiplier𝑃 =op(𝑝(𝑘)) with symbol𝑝 is said to be oforder 𝑟∈R if there exists a constant 𝐶 >0 such that

|𝑝(𝑘)| ≤ 𝐶^(︁1 +𝑘²^)︁^𝑟/2,

for every 𝑘 ∈ Z. In that case, for each 𝑞 ≥ 𝑟, the operator 𝑃 extends to a bounded linear operator from 𝐻^𝑞(S¹) to 𝐻^𝑞−𝑟(S¹).

2.2 Extending the Metric to 𝒟

^𝑞

( S

¹

)

Consider a 𝐿²-symmetric and positive definite Fourier multiplier𝐴 on C^∞(S¹). Fix 𝑟≥1, and suppose moreover, that

1. 𝑎(𝑘) =𝒪(|𝑘|^𝑟) : 𝐴 is oforder 𝑟.

2. For all𝑘∈Z,𝑎(𝑘)̸= 0 : 𝐴 isinvertible.

3. 1/𝑎(𝑘) =𝒪(|𝑘|^−𝑟) : 𝐴⁻¹ is oforder −𝑟.

In other words, 𝐴 extends, for all𝑞≥𝑟, to a topological isomorphism 𝐴:𝐻^𝑞(S¹)→𝐻^𝑞−𝑟(S¹)

for some fixed 𝑟≥1.

Example 2.3. Λ_2𝑠:=op^(︀(1 +𝑘²^)︀^𝑠) is of order 2𝑠, satisfies the requirements and corresponds to the inertia operator for the 𝐻^𝑠 Sobolev metric (𝑠≥1/2).

Remark 2.4. Notice that𝐴induces a bounded, injective linear operator from𝐻^𝑞(S¹) to its dual, but not a topological isomorphism.

The (smooth) right-invariant metric induced by 𝐴 on Diff^∞(S¹) is given on each tangent space by

⟨𝑣₁, 𝑣₂⟩_𝜙 = ˆ

S¹

𝑣₁(𝐴𝜙𝑣₂)𝜙𝑥𝑑𝑥.

and the flat map is represented by the operator 𝐴˜_𝜙 :=𝜙_𝑥𝐴_𝜙

It corresponds to an injective vector bundle morphism. Its image is the vector bundle𝒟^𝑞(S¹)× 𝐻^𝑞−𝑟(S¹) which maps continuously and one-to-one in the Hilbert bundle𝑇^*𝒟^𝑞(S¹)≈ 𝒟^𝑞(S¹)× 𝐻^𝑞(S¹). Notice however that𝒟^𝑞(S¹)×𝐻^𝑞−𝑟(S¹) is not a subbundle of𝑇^*𝒟^𝑞(S¹) in the sense of [31, III.3] because Λ_2𝑞(𝐻^𝑞−𝑟(S¹)) is not a closed subspace of 𝐻^𝑞(S¹).

The problem is that it is not at all obvious that the flat map ˜𝐴 and hence the metric is smooth. Indeed, the mapping

(𝜙, 𝑣)↦→𝑅_𝜙(𝑣), 𝒟^𝑞(S¹)×𝐻^𝑞(S¹)→𝐻^𝑞(S¹) is not differentiable(see [14] for instance).

When 𝐴 is a differential operator of order 𝑟, 𝐴_𝜙 is a linear differential operator whose coefficients are polynomial expressions of 1/𝜙𝑥 and the derivatives of 𝜙 up to order 𝑟 (e.g.

𝐷𝜙= (1/𝜙𝑥)𝐷). In that case, 𝜙↦→𝐴𝜙 is smooth (in fact real analytic) for𝑞 > 𝑟+ 1/2.

However for a general Fourier multiplier the general answer seems to be open. If we assume that𝐴is of finite order𝑟≥1, and choose 𝑞 > 𝑟+ 1/2, then𝐻^𝑞−𝑟(S¹) is a multiplicative algebra and since the map 𝜙↦→𝜙𝑥 is analytic, we are reduced to the following question.

(18)

Problem. Given a general Fourier multiplier 𝐴 of finite order 𝑟, under which condition is the mapping

𝜙↦→𝐴_𝜙:=𝑅_𝜙∘𝐴∘𝑅_𝜙⁻¹, 𝒟^𝑞(S¹)→ ℒ(𝐻^𝑞(S¹), 𝐻^𝑞−𝑟(S¹)) smooth for 𝑞 large enough?

Remark 2.5. In general, we cannot even conclude that the flat map is even continuous. Indeed, even if the mapping

(𝜙, 𝑣)↦→𝑅𝜙𝑣, 𝒟^𝑞(S¹)×𝐻^𝑞(S¹)→𝐻^𝑞(S¹) is continuous, this does not imply the the mapping

𝜙↦→𝑅_𝜙, 𝒟^𝑞(S¹)→ ℒ(𝐻^𝑞(S¹))

is continuous with respect to the operator norm on ℒ(𝐻^𝑞(S¹)) (norm continuity). In general, Norm continuity obviously implies continuity but the converse is false. Indeed, a general result in the theory of semigroups of linear operators states that a semigroup on a Banach space 𝐸 is norm continuous at 0 if and only if its infinitesimal generator is bounded on𝐸, cf. [37, Theorem 1.2]. Let now𝑞 >3/2 and let𝜏𝑠 be the rotation by the angle 𝑠on S¹. Then the representation of the group {𝑅_𝜏_𝑠;𝑠∈ R} is continuous on 𝐻^𝑞(S¹). But it cannot be norm continuous, since its infinitesimal generator 𝐷 is not bounded on 𝐻^𝑞(S¹). A direct argument, which shows that

‖𝑅_𝜏_𝑠−Id‖_ℒ(𝐻𝑞(S¹))is bounded away from 0 for all𝑠near 0 is runs as follows: Let𝑠∈(−1/2,1/2) and𝑢𝑠be a periodic, bump function with support in (𝑘−𝑠/2, 𝑘+𝑠/2) (𝑘∈Z) with‖𝑢_𝑠‖_𝐿2 = 1.

We have then

‖𝑅_𝜏_𝑠𝑢_𝑠−𝑢_𝑠‖²_𝐻𝑞(S¹)= 2‖𝑢_𝑠‖²_𝐻𝑞(S¹),

because 𝑢_𝑠 and 𝑅_𝜏_𝑠𝑢 are𝐻^𝑞(S¹)-orthogonal and 𝑅_𝜏_𝑠 is an𝐻^𝑞(S¹)-isometry. Hence

‖𝑅_𝜏_𝑠−Id‖_ℒ(𝐻𝑞(S¹))≥√

2 for −1

2 < 𝑠 < 1 2, which proves that the representation𝜙↦→𝑅𝜙 is not norm continuous.

The next two sections will deal with this difficult question.

2.3 A Necessary and Sufficient Condition for Smoothness

A Necessary Condition

Let A a Fourier multiplier and suppose that the map

𝜙↦→𝐴_𝜙 :=𝑅_𝜙∘𝐴∘𝑅_𝜙⁻¹, 𝒟^𝑞(S¹)→ ℒ(𝐻^𝑞(S¹), 𝐻^𝑞−𝑟(S¹)) is smooth¹. Then we can compute its Fréchet differential

𝜕𝜙𝐴𝜙∈ ℒ²(𝐻^𝑞(S¹), 𝐻^𝑞−𝑟(S¹)),

which is itself smooth. Inductively, we define this way its𝑛-th Fréchet differential

𝜕_𝜙^𝑛𝐴_𝜙 ∈ ℒ^𝑛+1(𝐻^𝑞(S¹), 𝐻^𝑞−𝑟(S¹))

1We have chosen to denote by𝜕 the Fréchet differential to avoid the confusion with the already used notation 𝐷=𝑑/𝑑𝑥.

Geodesic Flows on the Diffeomorphism Group of the Circle