Geodesic flow on the diffeomorphism group of the circle

Transkript

1 Comment. Math. Helv. 78 (2003) /03/ DOI /s c 2003 Birhäuser Verlag, Basel Commentarii Mathematici Helvetici Geodesic flow on the diffeomorphism group of the circle Adrian Constantin and Boris Kolev Abstract. We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds. Mathematics Subject Classification (2000). 35Q35, 58B25. Keywords. Geodesic flow, diffeomorphism group of the circle. 1. Introduction The group D of all smooth orientation-preserving diffeomorphisms of the circle S is the simplest possible example of an infinite-dimensional Lie group [1]. Its Lie algebra T Id D is the space C (S) of the real smooth periodic maps of period one. Since C (S) is not provided with a natural inner product, to endow D with a Riemannian structure we have to define an inner product on each tangent space T η D, η D. For a Lie group the Riemannian exponential map of any two-sided invariant metric coincides with the Lie group exponential map [1]. It turns out that the Lie group exponential map on D is not locally surjective cf. [12] so that a meaningful 1 Riemannian structure cannot be provided by a bi-invariant metric on D. We are led to define an inner product on C (S) and produce a right-invariant metric 2 by transporting this inner product to all tangent spaces T η D, η D,by means of right translations. In this paper we show that certain right-invariant metrics induce noteworthy Riemannian structures on D. Despite the analytical difficulties that are inher- 1 A prerequisite of a rigorous study aimed at proving infinite-dimensional counterparts of facts established in classical (finite-dimensional) Riemannian geometry is the use of the Riemannian exponential map as a local chart on D. 2 To carry out the passage from right-invariant metrics to left-invariant ones, note that a right-invariant metric on D is transformed by the inverse group operation to a left-invariant metric on D with the reverse law (ϕ ψ= ψ ϕ).

2 788 A. Constantin and B. Kolev CMH ent 3, the existence of geodesics is obtained and their length-minimizing property is established. The paper is organized as follows. In Section 2 we discuss the manifold and Lie group structure of D, Section 3 is devoted to basic properties of the Riemannian structures that we construct on D (existence and local chart property of the Riemannian exponential map), while in Section 4 we prove the length-minimizing property of geodesics. In the last section we present 4 a choice of a right-invariant matric endowing D with a deficient Riemannian structure (the corresponding Riemannian exponential map is not a local C 1 -diffeomorphism), to emphasize the special features of the previously discussed Riemannian structures. Note that for diffeomorphism groups the existence of geodesics is an open question 5 so that it is of interest to have an example (M = S) where an attractive geometrical structure is available. 2. The diffeomorphism group In this section we discuss the manifold and Lie group structure of D. If ξ(x) is a tangent vector to the unit circle S at x S C, then R [x ξ(x)] = 0 and u(x) = 1 xξ(x) R. 2πi This allows us to identify the space of smooth vector fields on the circle with C (S), the space of real smooth maps of the circle. The latter may be thought of as the space of real smooth periodic maps of period one and will be used as a model for the construction of local charts on D. Note that C (S) isafréchet space, its topology being defined by the countable collection of C n (S)-seminorms: a sequence u j u if and only if for all n 0wehaveu j u in C n (S) asj. D is an open subset of C (S; S) C (S; C), as one can easily see considering the function defined on C (S; S) by Θ(ϕ) = inf x y ϕ(x) ϕ(y). x y We will describe a Fréchet manifold structure on D. If t ϕ(t) isac 1 -path in D with ϕ(0) = Id,wehaveϕ (0)(x) T x S. Therefore ϕ (0) is a vector field on S and we can identify T Id D with C (S). If ϕ Dis such that ϕ Id C 0 (S) < 1/2, 3 D is a Fréchet manifold so that the inverse function theorem and the classical local existence theorem for differential equations with smooth right-hand side are not granted cf. [12]. Moreover, we deal with wea Riemannian metrics (the family of open sets of D contains but does not coincide with the family of open sets of the topology induced by the metric) so that even the existence of a covariant derivative associated with the right-invariant metric is in doubt cf. [15]. 4 For a detailed analysis see [7]. 5 For any smooth compact manifold M, both the group of smooth diffeomorphisms of M, Diff(M), and its subgroup formed by the volume-preserving diffeomorphisms, have a Lie group structure [16]. Progress towards the existence of geodesics was made but the understanding is still incomplete [1].

3 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 789 we can define u(x) = 1 ( ) 2πi Log xϕ(x) C (S; R). Note that u(x) is a measure of the angle between x and ϕ(x). Define a lift F ϕ : R R of ϕ such that u Π( x) =F ϕ ( x) x, x R, where Π : R S is the cover map. In the neighborhood U ϕ0 = { ϕ ϕ 0 C0 (S) < 1/2} of ϕ 0 Dwe are led to define u(x) = 1 ( ) 2πi Log ϕ 0 (x) ϕ(x), x S, and u Π( x) =F ϕ ( x) F ϕ0 ( x), x R. For ϕ U ϕ0, let Ψ ϕ0 (ϕ) =u. We obtain the local charts {U ϕ0, Ψ ϕ0 }, with the change of charts given by 6 Ψ ϕ2 Ψ 1 ϕ 1 (u 1 )=u πi Log(ϕ 2 ϕ 1 ). The previous transformation being just a translation on the vector space C (S), the structure described above endows D with a smooth manifold structure based on the Fréchet space C (S). A direct computation shows that the composition and the inverse are both smooth maps from D Dto D, respectively from D to D, so that the group D is a Lie group. Note that the derivatives of the left-translation L η : D D, L η (ϕ) =η ϕ, η D, and right-translation R η : D D, R η (ϕ) =ϕ η, η D, are given by L η : T Id D T η D, u η x u, η D, respectively R η : T Id D T η D, u u η, η D. The Lie bracet on the Lie algebra T Id D C (S) ofd is [u, v] = (u x v uv x ), u,v C (S). (2.1) 6 With u 1 =Ψ ϕ1 (ϕ) andu 2 =Ψ ϕ2 (ϕ) forϕ U ϕ1 U ϕ2,wehave (2πi) u 2 = Log(ϕ 2 ϕ)=log(ϕ 2 ϕ 1 ϕ 1 ϕ)=log(ϕ 2 ϕ 1 ) + Log(ϕ 1 ϕ)=log(ϕ 2 ϕ 1 )+(2πi) u 1. Hence Ψ ϕ2 (ϕ) =Ψ ϕ1 (ϕ)+ 1 2πi Log(ϕ 2 ϕ 1 ) and the change of charts is plain.

4 790 A. Constantin and B. Kolev CMH Each v T Id D gives rise to a one-parameter group of diffeomorphisms {η(t, )} obtained solving η t = v(η) in C (S) (2.2) with data η(0) = Id D. Conversely, each one-parameter subgroup t η(t) D is determined by its infinitesimal generator v = t η(t) t=0 T Id D. Evaluating the flow t η(t, ) of (2.2) at t = 1 we obtain an element exp L (v) ofd. The Lie-group exponential map v exp L (v) is a smooth map of the Lie algebra to the Lie group. Although the derivative of exp L at 0 C (S) is the identity, exp L is not locally surjective cf. [16]. This failure, in contrast with the case of Hilbert manifolds (see [15]), is due to the fact that the inverse function theorem does not necessarily hold in Fréchet spaces cf. [12]. This inconvenience might seem to indicate that it is preferable to wor with the Hilbert manifolds 7 D = {η H (S) is bijective, orientation-preserving, and η 1 H (S)}, 2, instead of D. However, D is only a topological group and not a Lie group since for η D the right composition R η : D D is smooth but both the left composition L η and the inverse map ϕ ϕ 1 are merely continuous on D, without being smooth (see [2]). Therefore, in order to obtain a Lie group structure, we have to consider the Fréchet manifold D. Let F(D) be the ring of smooth real-valued functions defined on D and X (D) be the F(D)-module of smooth vector fields on D. ForX X(D) andf F(D), the Lie derivative L X f is defined in a local chart as f(ϕ + hx(ϕ)) f(ϕ) L X f(ϕ) = lim, ϕ D. h 0 h If U Dis open and X, Y : U C (S) are smooth, let Y (ϕ + hx(ϕ)) Y (ϕ) D X Y (ϕ) = lim, ϕ D. h 0 h This leads to a covariant definition of the Lie bracet of X, Y X(D), L X Y =[X, Y ]=D X Y D Y X. Note that if X R (D) is the space of all right-invariant smooth vector fields 8 on D, then the bracet [X, Y ]ofx, Y X R (D) is a right-invariant vector field and 7 H (S), 0, is the space of all L 2 (S)-functions (square integrable functions) f with distributional derivatives up to order, x i f with i =0,...,,inL2 (S). Endowed with the norm f 2 = ( xf) i 2 (x) dx, i=0 S H (S) becomes a Hilbert space. Note that if { ˆf(j)} j Z is the Fourier series of f H (S), then f 2 = ( 1+(2πj) (2πj) 2) ˆf(j) 2. j Z 8 X X R (D) is determined by its value u at Id, X η = R η u for η D.

5 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 791 [X, Y ] Id =[u, v], where u = X Id, v = Y Id cf. [16]. This feature explains the minus sign entering formula (2.1) the commutation operation is defined by this construction carried out with right-invariant vector fields. 3. Riemannian structures on D We define an inner product on the Lie algebra T Id D C (D) ofd, and extend it to D by right-translation. The resulting right-invariant metric on D will be a wea Riemannian metric. In this section we discuss the existence of the geodesic flow associated with this metric. Consider on T Id D C (S) theh (S) inner product u, v = ( xu)( i xv) i dx, u, v C (S), i=0 S and extend this inner product to each tangent space T η D, η D, by righttranslation, i.e. V,W := R η 1 V, R η 1 W, V,W T ηd. (3.1) We have thus endowed D with a smooth right-invariant metric. Note that the right-invariant metric (3.1) defines a wea topology on D so that the existence of a covariant derivative which preserves the inner product (3.1) is not ensured on general grounds cf. [15]. We will give a constructive proof of the existence of such a covariant derivative. Let us first note that u, v = A (u) v dx, u, v H (S), 0, S where for every n 0, A : H n+2 (S) H n (S) is the linear continuous isomorphism A =1 d2 d2 + +( 1). (3.2) dx2 dx2 This enables us to define the bilinear operator B : C (S) C (S) C (S), ( ) B (u, v) = A 1 2v x A (u)+va (u x ), u,v C (S), (3.3) with the property that B (u, v), w = u, [v, w], u,v,w C (S). We can extend B to a bilinear map B on the space X R (D) of smooth rightinvariant vector fields on D by B (X, Y ) η = R η B (X Id,Y Id ), η D,X,Y X R (D).

6 792 A. Constantin and B. Kolev CMH For X X(D), let us denote by Xη R whose value at η is X η. the smooth right-invariant vector field on D Theorem 1. Let 0. There exists a unique Riemannian connection on D associated to the right-invariant metric (3.1), with ( XY ) η =[X, Y Yη R ] η + 1 ( [Xη R,Yη R ] η B (Xη R,Yη R ) η B (Yη R,Xη R ) η ), 2 for smooth vector fields X, Y on D. Proof. The uniqueness of is obtained lie in classical Riemannian geometry (see e.g. [15]) and all the required properties can be checed from its explicit representation, using the defining identity for B. The existence of enables us to define parallel translation along a curve on D and to derive the geodesic equation of the metric defined by (3.1). Throughout the discussion, let J R be an open interval with 0 J. ForaC 1 -curve α : J D, let Lift(α) be the set of lifts of α to T D. The derivation D αt :Lift(α) Lift(α) along α is given in local coordinates by D αt γ = γ t Q (α t α 1,γ α 1 ) α, γ Lift(α), (3.4) where Q : C (S) C (S) C (S) is the bilinear operator Q (u, v) = 1 ( ) u x v + uv x + B (u, v)+b (v, u), 2 u,v C (S). For a C 1 -curve α : J Dwe have d dt γ 1,γ 2 = D αt γ 1,γ 2 + γ 1,D αt γ 2, t J, (3.5) for all γ 1,γ 2 Lift(α). If α : J Dis a C 2 -curve, a lift γ : J T D is called α-parallel if D αt γ 0 on J. This is equivalent to requiring that v t = 1 ( ) vu x v x u + B (u, v)+b (v, u), 2 (3.6) where u, v C 1 (J; C (S)) are defined as α t α 1 = u, respectively γ α 1 = v. A C 2 -curve ϕ : J D satisfying D ϕt ϕ t 0onJ is called a geodesic. If u = ϕ t ϕ 1 T Id D C (S), then u satisfies the equation u t = B (u, u), t J. (3.7) Equation (3.7), called the Euler equation, is the geodesic equation transported by right-translation to the Lie algebra T Id D. Equations of type (3.7) arise in mathematical physics.

7 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 793 Example 1. For = 0, that is, for the L 2 right-invariant metric, equation (3.7) becomes the inviscid Burgers equation u t +3uu x =0. This equation is a simplified model for the occurence of shoc waves in gas dynamics and can be studied quite explicitly [13]: all solutions but the constant functions have a finite life span. Example 2. For =1,thatis,fortheH 1 right-invariant metric, equation (3.7) becomes cf. [17] the Camassa Holm equation u t + uu x + x (1 x 2) 1 ( u u x 2 ) =0. This equation is a model for the unidirectional propagation of shallow water waves [3] as well as a model for axially symmetric waves on hyperelastic rods [9]. It has a bi-hamiltonian structure [11] and is completely integrable [8]. Some solutions of the Camassa Holm equation exist globally in time [5], whereas others develop singularities in finite time [6]. The blowup phenomenon can be interpreted as a simplified model for wave/rod breaing the solution (representing the surface water wave or the deformation of the hyperelastic rod) stays bounded while its slope becomes vertical in finite time [4]. In a local chart the geodesic equation (3.7) is ϕ tt = P (ϕ, ϕ t ), where P is an operator that will be specified in the proof of Theorem 2. Assuming for the moment the local existence of geodesics on D for the metric (3.1), proved below, let us derive a conservation law for the geodesic flow 9. Observe that any v C (S) T Id D defines a one-parameter group of diffeomorphisms h s : D D, h s (ϕ) = ϕ exp L (sv), where exp L is the Lie-group exponential map. The metric being by construction invariant under the action of h s, Noether s theorem ensures that if g : T D R stands for the right-invariant metric, then g [ dh s (ϕ) ] (ϕ, ϕ t ) ϕ t ds s=0 is preserved along the geodesic curve t ϕ(t) with ϕ(0) = Id and ϕ t (0) = u 0 T Id D. We compute dh s (ϕ) g = ϕ x v, ds s=0 v (ϕ, v)[w] =2 v ϕ 1,w ϕ 1, 9 For finite-dimensional Lie groups the geodesic flow of a one-sided invariant metric the angular momentum is preserved [1]. This is a consequence of the invariance of the metric by the action of the group on itself, in view of Noether s theorem. The same reasoning can be carried over to the present infinite-dimensional case.

8 794 A. Constantin and B. Kolev CMH obtaining that Therefore S ϕ t ϕ 1,ϕ x ϕ 1 v ϕ 1 = u 0,v, v C (S). A (u) ϕ x ϕ 1 v ϕ 1 dx = A (u 0 ) vdx, S v C (S), where, as before, u = ϕ t ϕ 1. A change of variables yields A (u) ϕ ϕ 2 x vdx= A (u 0 ) vdx, v C (S) S so that, denoting we obtain S m = A (u) ϕ ϕ 2 x, (3.8) m (t) =m (0), t [0,T), (3.9) where ϕ C 2 ([0,T); D) is the geodesic for the metric (3.1), starting at ϕ(0) = Id Din the direction u 0 = ϕ t (0) T Id D; as before, u = ϕ t ϕ 1. To prove the existence of geodesics, we proceed as follows. The classical local existence theorem for differential equations with smooth right-hand side, valid for Hilbert spaces (see [15]), does not hold in C (S) cf. [12]. However, note that C (S) = n 2+1 H n (S). We use the classical approach to prove that for every n 2 + 1, the geodesic equation has, on some maximal interval [0,T n ) with T n > 0, a unique solution in H n (S), depending smoothly on time. A priori T n T 2+1. It turns out that T n = T 2+1 for all n 2 +1, a fact that will ensure the existence of geodesics on D endowed with the right-invariant metric (3.1) for every 1. The peculiarities of the special case = 0 (where this approach is not applicable) are discussed in the last section. Theorem 2. Let 1. For every u 0 C (S), there exists a unique geodesic ϕ C ([0,T); D) for the metric (3.1), starting at ϕ(0) = Id Din the direction u 0 = ϕ t (0) T Id D. Moreover, the solution depends smoothly on the initial data u 0 C (S). Proof. Note that ua (u x )=A (uu x )+C 0 (u), u H n (S), n 2 +1, where C 0 : Hn (S) H n 2 (S) isac -operator depending quadratically on u, u x,..., x 2 u. Denoting by C : H n (S) H n 2 (S) thec -operator C (u) = C(u) 0 2u x A (u), we obtain that B (u, u) =A 1 C (u) uu x, u H n (S), n 2 +1.

9 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 795 The geodesic equation (3.7) becomes u t + uu x = A 1 C (u), where u = ϕ t ϕ 1 C (S). Letting v = u ϕ = ϕ t, we can write the geodesic equation in a local chart U in C (S) as { ϕ t = v, (3.10) v t = P (ϕ, v), where The operator P (ϕ, v) = [ ] A 1 C (v ϕ 1 ) ϕ. (ϕ, v) (ϕ, P (ϕ, v)) can be decomposed into Q E with ( ) E (ϕ, v) = ϕ, R ϕ C R ϕ 1(v), and Q (ϕ, v) = ( ) ϕ, R ϕ A 1 R ϕ 1(v). Specifying the explicit form of E (ϕ, v), we see that this operator extends to the space U n H n (S), where U n is the open subset of H n (S) of all functions having a strictly positive derivative 10. The same argument can be pursued in the case of the operator G : U n H n (S) U n H n 2 (S), ( ) G (ϕ, v) = ϕ, R ϕ A R ϕ 1(v), the inverse of Q (as a map). Direct calculations confirm that E and G are both smooth maps from U n H n (S) tou n H n 2 (S). The regularity of G ensures that its Fréchet differential can be computed by calculating directional derivatives. One finds that ( ) Id 0 DG (ϕ, v) = 2 i=0 a i(ϕ) x i, with a 2 = ( 1) ϕ 2 x and a 0,...,a 2 1, all of the form p (ϕ, ϕ x,..., 2 x ϕ) ϕ 4 x 10 Note that n sothatH n (S)-functions are of class C 2. Explicit calculations show that if η U n,thenη is a H n (S)-homeomorphism of the circle with η 1 H n (S).

10 796 A. Constantin and B. Kolev CMH for a polynomial p with constant coefficients, while is a linear differential operator of order 2 with coefficients rational functions of the form q(ϕ, v, ϕ x,v x,..., x 2 ϕ, x 2 v) ϕ 4 x for some constant coefficient polynomial q. Observe 11 that for every f H n 2 (S) there is a unique solution u H n (S) of the ordinary linear differential equation with H n 2 (S)-coefficients 2 a i (ϕ) xv i = f. i=0 Taing into account the form of DG, we infer that ( ) DG (ϕ, v) Isom U n H n (S), U n H n 2 (S). Since the differential of the smooth map G : U n H n (S) U n H n 2 (S) is invertible at every point, from the inverse function theorem on Hilbert spaces [15] we deduce that its inverse Q : U n H n 2 (S) U n H n (S) is also smooth. The regularity properties that we just proved for the maps Q, E, show that P is a smooth map from U n H n (S) toh n (S). Regard (3.10) as an ordinary differential equation on U n H n (S), with a smooth right-hand side, viewed as a map from U n H n (S) tou n H n (S). The Lipschitz theorem for differential equations in Banach spaces [15] ensures that for every ball B(0,ε n ) H n (S) there exists T n = T n (ε n ) > 0 such that for every u 0 B(0,ε n ), the equation (3.10) with data ϕ(0) = Id and v(0) = u 0 has a unique solution (ϕ, v) C ([0,T n ); U n H n (S)). Moreover, this solution (ϕ, v) depends smoothly on the initial data u 0 and can be extended to some maximal existence time Tn > 0. If T n <, we have either that lim sup t T n v(t) n = or there is a sequence t j T n such that ϕ(t j ) accumulates at the boundary of U n as j. Choose some ball B(0,ε 2+1 ) H 2+1 (S). We prove now that for any u 0 B(0,ε 2+1 ) C (S) there exists a unique geodesic ϕ C ([0,T 2+1 ); D) for the metric (3.1), starting at ϕ(0) = Id in the direction u 0. Since u 0 H n (S) for every n 2 + 1, it suffices to prove that the solution (ϕ, v) of equation (3.10) on each U n H n (S), with data ϕ(0) = Id and v(0) = u 0, has the maximal existence time T n = T 2+1. Assuming that T 2+2 <T 2+1, note that (ϕ(t 2+2 ),v(t 2+2 )) is defined in U 2+1 H 2+1 (S) andϕ(t 2+2 )isac 1 -diffeomorphism of the circle. Recall the notation u = v ϕ 1. To prove that ϕ(t) converges in U 2+2 (S) ast T 2+2, let us use ϕ t = u ϕ to compute x 2 ϕ t, t (0,T 2+1 ). We obtain ϕ x 2 x ϕ t ϕ tx 2 x ϕ =( 1) ϕ 2 x [ ϕ 2 3 x m (t)+e (v, ϕ) ], t (0,T 2+1 ), 11 This can be proved using the Fourier representation of functions in H j (S), j 0.

11 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 797 where E (v, ϕ) is a smooth expression containing only x-derivatives of ϕ of order i 2 1andx-derivatives of v of order j 2 1. Hence d ( 2 x ϕ ) [ ] =( 1) ϕ 2 3 x m (t)+e (v, ϕ) dt ϕ x =( 1) [ ϕ 2 3 x m (0) + E (v, ϕ) ], t (0,T 2+1 ), in view 12 of (3.9). For t (0,T 2+1 ) we obtain that t [ ] x 2 ϕ(t) =( 1) ϕ x (t) ϕ 2 3 x m (0) + E (v, ϕ) ds. (3.11) Since m (0) C (S) and 0 (ϕ, v) C ([0,T 2+1 ); U 2+1 H 2+1 (S)) C ([0,T 2+2 ); U 2+2 H 2+2 (S)), differentiating (3.11) twice with respect to x, we infer that (ϕ(t),ϕ t (t)) converges in U 2+2 (S) H 2+2 (S) ast T 2+2. The limit can only be (ϕ(t 2+2 ),v(t 2+2 )). Therefore T 2+2 = T 2+1. This procedure can be repeated for n =2 + 3 etc. and the existence of the smooth geodesics on D is now plain. The previous results enable us to define the Riemannian exponential map exp for the H right-invariant metric ( 1). If ϕ(t; u 0 ) is the geodesic on D, starting at Id in the direction u 0 C (S), note the homogeneity property ϕ(t; su 0 )=ϕ(ts; u 0 ) (3.12) valid for all t, s 0 such that both sides are well-defined. In the proof of Theorem 2 we saw that there exists δ>0sothat all geodesics ϕ(t; u 0 ) are defined on the same time interval [0,T] with T>0, for all u 0 Dwith u <δ. Hence, we can define exp(u 0 )=ϕ(1; u 0 ) on the open set { u 0 D: u < 2 δ } T of D, and the map u 0 exp(u 0 ) is smooth. Theorem 3. The Riemannian exponential map for the H right-invariant metric on D, 1, is a smooth local diffeomorphism from a neighborhood of zero on T Id D to a neighborhood of Id on D. Let us first establish 12 Relation (3.9) was derived assuming the existence of geodesics on D. In the present context, it can be proved as follows. Define m by (3.8) and note that it is a polynomial expression in v, ϕ, v x,ϕ x,..., x 2 v, x 2 ϕ, divided by some power of ϕ x. Therefore m C ([0,T 2+1 ); H 1 (S)) and, using (3.10), it can be checed by differentiation with respect to t that m (t) =m (0) for all t [0,T 2+1 ).

12 798 A. Constantin and B. Kolev CMH Lemma 1. Let n 2+1 and let (ϕ, v) be a solution of (3.10) with data (Id,u 0 ) U n H n (S), definedon[0,t). If there exists t [0,T) such that ϕ(t) U n+1 then u 0 H n+1 (S). Proof. From (3.11) we get x 2 ϕ(t) =( 1) m (0) ϕ x (t) t 0 ϕ 2 3 x t ds +( 1) E (v, ϕ) ds, 0 t [0,T). If for some t [0,T)wehaveϕ(t) H n+1 (S), the fact that ϕ x is strictly positive forces m (0) H n+1 2 (S) and therefore u 0 H n+1 (S). Proof of Theorem 3. Viewing exp as a smooth map from a small neighborhood of 0 H n (S) tou n, n 2 + 1, its differential at 0 H n (S), Dexp 0, is the identity map. Indeed, for v H n (S) we have by (3.12) that exp(tv) =ϕ(t; v) so that d dt exp(tv) t=0 = d dt ϕ(t; v) t=0 = v. As a consequence of the inverse function theorem on Hilbert spaces, we can find open neighborhoods V 2+1 and O 2+1 of 0 H 2+1 (S) andid U 2+1, respectively, such that exp : V 2+1 O 2+1 is a C diffeomorphism with Dexp u0 : H 2+1 (S) H 2+1 (S) bijective for every u 0 V 2+1. We already now from the proof of Theorem 2 that ( ) exp V 2+1 C (S) O 2+1 C (S), while Lemma 1 ensures that exp is a local bijection between these open sets. It remains to show that exp is a smooth diffeomorphism from V 2+1 C (S) to O 2+1 C (S). Let u 0 V 2+1 C (S). We now that Dexp u0 is a bounded linear operator from H n (S) toh n (S) for every n and we will prove that it is actually a bijection. Then, in view of the inverse function theorem on Hilbert spaces, both exp and its inverse are smooth maps on small H n (S)-neighborhoods of u 0 V 2+1 C (S), respectively exp(u 0 ) O 2+1 C (S). Letting n, this would show that exp is locally a smooth diffeomorphism. To prove this last step, we use an inductive argument. To start with, Dexp u0 is a bijection from H 2+1 (S)toH 2+1 (S)asu 0 V 2+1.Forafixedn 2+1, assume that the map Dexp u0 is a bijection from H j (S) toh j (S) for all j =2 +1,...,n, and let us show that Dexp u0 is a bijection from H n+1 (S) toh n+1 (S). First of all, Dexp u0 is injective as a bounded linear map from H n+1 (S) toh n+1 (S) since its extension to H n (S) is injective. To prove its surjectivity as a map from H n+1 (S) to H n+1 (S), it suffices 13 to see that there is no w H n (S), w H n+1 (S), with Dexp u0 (w) H n+1 (S). Assume there is such a w. Forε>0small enough let ϕ ε (t) 13 Since exp is a smooth map on V 2+1 H n+1 (S) wehavedexp u0 (H n+1 (S)) H n+1 (S) while the inductive assumption ensures Dexp u0 (H n (S)) = H n (S).

13 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 799 be the solution of (3.10) on U n starting at Id in the direction u 0 + εw, with the corresponding v ε H n (S)). We now that the map (ϕ ε (t),v ε (t)) U n H n (S) depends smoothly on ε and t [0, 1]. From (3.11) we obtain ( 1) 2 x ϕ ε (1) ( ) 1 1 ϕ ε = m (0; u 0 )+εm (0; w) (ϕ ε x) 2 3 ds + E (v ε,ϕ ε ) ds. x(1) 0 0 Differentiating with respect to ε, a calculation shows that Dexp u0 (w)= d dε ϕε (1) H n+1 (S) is possible only if m (0; w) H n 2+1 (S), i.e. w H n+1 (S). The obtained contradiction concludes the proof. ε=0 4. Minimizing property of the geodesics Throughout this section we prove the length minimizing property for the geodesics of the right-invariant metric (3.1) on D for some fixed 1. Let V 0 be a vector tangent at α(0) = α 0 to a C 2 -curve α : J D. The parallel transport of V along the curve α is defined as a curve γ Lift(α) with γ(0) = V 0 and D αt γ 0onJ. Lemma 2. Let α : J Dbe a C 2 curve. Given V 0 T α0 D,α 0 = α(0) D, there exists a unique lift γ : J T D which is α-parallel and such that γ(0) = V 0. Moreover, if γ 1,γ 2 are the unique α-parallel lifts of α with γ i (0) = V i T α0 D,i= 1, 2, then γ 1 (t), γ 2 (t) = V 1,V 2, t J. Proof. In view of (3.3) and (3.6), the equation of parallel transport is v t = 1 [ 2 (vu x uv x ) A 1 v x A (u)+u x A (v)+ 1 2 va (u x )+ 1 ] 2 ua (v x ), where u = α t α 1 and v = γ α 1. Note that the operators (u, v) A 1 [v xa (u)+u x A (v)], (u, v) 1 [ ] 2 A 1 va (u x )+ua (v x ) 1 2 (vu x + uv x ) are smooth from H n (S) H n (S) toh n (S) for every n 2+1. Denote by Θ (u, v) their sum. The equation of parallel transport can be written as v t + uv x +Θ (u, v) =0. (4.1) For a fixed u C 1 (J; D), the map v Θ (u, v) is a bounded linear operator from H n (S) toh n (S) for every n Viewing (4.1) as linear hyperbolic evolution equation in v with fixed u C 1 (J; D), it is nown (see [14]) that, given V 0 H n (S), n 2 + 1, there exists a unique solution v C(J; H n (S)) C 1 (J; H n 1 (S))

14 800 A. Constantin and B. Kolev CMH of (4.1) with initial data v(0) = V 0. Letting n, we infer that, given V 0 α0 1 T Id D C (S), there exists a unique solution v C 1 (J; D) to (4.1) with v(0) = V 0 α0 1. From (3.5) we deduce that γ 1 (t), γ 2 (t) is constant for any α-parallel lifts and the second assertion follows. Choose open neighborhoods W of 0 C (S), respectively U of Id D, such that Dexp u0 : H 2+1 (S) H 2+1 (S) is bijective for every u 0 Wand exp is a smooth diffeomorphism from W onto U, cf. Theorem 3. The map ( ) G : D W D D, (η, u) η, R η exp(u), is a smooth diffeomorphism onto its image. Let U(η) =R η U = R η exp(w). If ϕ U(η) {η}, then ϕ = exp(v) η for some v W. Letv = rw, where w, w =1andr R + to define the polar coordinates (r, w) ofϕ U(η). If σ : J 1 J 2 Dis a map such that 2 σ r 2, 2 σ t r and 2 σ are continuous, r t denote by 1 σ the partial derivative with respect to r and define similarly 2 σ. Both curves r 1 σ(r, t) andr 2 σ(r, t) are lifts of r σ(r, t). Generally, if γ is a lift of r σ(r, t), let (D 1 γ)(r, t) =(D 1σγ)(r) and define D 2 γ similarly. In a local chart we have by (3.4) that D 1 2 σ = 1 2 σ Q ( 1 σ, 2 σ)=d 2 1 σ (4.2) since Q is symmetric. On the other hand, from (3.5) we infer 2 1 σ, 1 σ =2 D 2 1 σ, 1 σ. The previous relation combined with (4.2) yields 2 1 σ, 1 σ =2 D 1 2 σ, 1 σ. (4.3) Lemma 3. Let γ :[a, b] U(η) {η} be a piecewise C 1 -curve. Then l(γ) r(b) r(a), where l(γ) is the length of the curve and (r(t),w(t)) are the polar coordinates of γ(t). Equality holds if and only if the function t r(t) is monotone and the map t w(t) W is constant. Proof. We may assume without loss of generality that γ is C 1 (in the general case, brea γ up into pieces that are C 1 ) and that η = Id (in view of the right-invariance property of the metric). Observe that w(t) is obtained in a chart by the inversion of exp followed by a projection so that the functions t r(t) andt w(t) areof class C 1. Let σ(r, t) =exp(rw(t)). Let ϕ(s; z) be the solution of (3.10) starting at Id in the direction z C (S). Relation (3.12) yields σ(r, t) =ϕ(r; w(t)), while the proof of Theorem 2 ensures for every n the smooth dependence of ϕ(s; z)

15 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 801 on s as well as the smooth dependence of (ϕ, ϕ s )onz in H n (S). Therefore 2 σ r 2 and 2 σ t r are continuous in the Hn (S)-setting for every n Furthermore, since s ϕ ϕ(s; z) =Id + s (ξ; z) dξ in Hn (S), 0 we have ϕ s 2 ϕ (s; z) = z 0 z s (ξ; z) dξ in L(Hn (S),H n (S)), thus 2 ϕ z s = 2 ϕ s z. But t w(t) Hn (S) isac 1 2 σ -map so that is also r t continuous in the H n (S)-setting for every n Letting n we obtain that 2 σ r 2, 2 σ r t and 2 σ t r are all continuous in the C (S)-topology. Note that γ (t) = σ r r (t) + σ, t t J. (4.4) Since r σ(r, t) is a geodesic, we obtain by Lemma 2 that σ r, σ = w(t),w(t) 1. r (4.5) Let us now show that σ r, σ 0. t (4.6) Indeed, from (4.3) and (4.5) we obtain that σ D 1 t, σ = 1 σ r 2 t r, σ 0. r This, in combination with (3.5), leads to σ r r, σ σ = D 1 t r, σ σ + t r,d σ 1 0, t σ since (D 1 r )=0asr σ(r, t) is a geodesic. The previous relation yields σ r, σ σ (r, t) = t r, σ (0,t). t But σ(0,t)=id forces σ (0,t) = 0 and therefore (4.6) holds. r Combining (4.4) (4.6), we obtain 2 γ (t) 2 = r (t) 2 + σ t r (t) 2, t [a, b],

16 802 A. Constantin and B. Kolev CMH so that the length of γ is estimated by l(γ) b a r (t) dt r(b) r(a). Since σ t 0 forces w (t) =0asDexp rw(t) is a bijection from H 2+1 (S) to H 2+1 (S), the characterization of the equality case follows at once. Let us now prove Theorem 4. If η, ϕ Dare close enough, more precisely, if ϕ η 1 U, then η and ϕ can be joined by a unique geodesic in U(η). Among all piecewise C 1 -curves joining η to ϕ on D, the geodesic is length minimizing. Proof. Observe that if v = exp 1 (ϕ η 1 ), then α(t) =exp(tv) η is the unique geodesic joining η to ϕ in U(η) cf. Theorem 3. To prove the second statement, let ϕ η 1 = exp(rw) with w = 1 and choose ε (0,r). If γ is any piecewise C 1 -curve on D joining η to ϕ, then γ contains an arc of curve γ such that, after reparametrization, and exp 1 (γ (0)) = ε, exp 1 (γ (1)) = r, ε exp 1 (γ (t)) r, t [0, 1]. Lemma 3 yields l(γ ) r ε, thusl(γ) l(γ ) r ε. The arbitraryness of ε>0 ensures l(γ) r. But l(α) =r in view of Lemma 3 and the minimum is attained if and only if the curve is a reparametrization of a geodesic. Remar. Specializing = 1 in Theorem 4 we obtain that for the Camassa Holm model (Example 2 in Section 3) the Least Action Principle holds. That is, a state of the system is transformed to another nearby state through a uniquely determined flow that minimizes the inetic energy cf. [7]. 5. Comments This section is devoted to a discussion of the L 2 (S) right-invariant metric on D, case when the geodesic equation (3.7) is the inviscid Burgers equation u t +3uu x =0 cf. [1]. The crucial difference from the case of a H (S) right-invariant metric (with 1) lies in the fact that the inverse of the operator A, defined by (3.2), is not regularizing. This feature maes the previous approach inapplicable but the existence of geodesics can be proved by the method of characteristics.

17 Vol. 78 (2003) Geodesic flow on the diffeomorphism group of the circle 803 Proposition 1 [7]. For the L 2 (S) right-invariant metric on D there exists a unique smooth geodesic on D starting at Id in the direction u 0 T Id D. This result enables one to define the Riemannian exponential map of the L 2 (S) right-invariant metric on D, in analogy to the cases considered in the present paper. However, Proposition 2 [7]. The Riemannian exponential map of the L 2 (S) right-invariant metric on D is not a C 1 -diffeomorphism from a neighborhood of zero in T Id D C (S) to a neighborhood of the identity on D. The question whether another right-invariant metric could provide D with a nice Riemannian structure has been positively answered in this paper. Acnowledgements. We than L. Hörmander for his helpful interest and V. Schroeder for stimulating conversations. References [1] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New Yor, [2] H. Brézis and J. P. Bourguignon, Remars on the Euler equation, J. Funct. Anal. 15 (1974), [3] R. Camassa and D. Holm, An integrable shallow water equation with peaed solitons, Phys. Rev. Letters 71 (1993), [4] A. Constantin and J. Escher, Wave breaing for nonlinear nonlocal shallow water equations, Acta Mathematica 181 (1998), [5] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998), [6] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaing waves for a shallow water equation, Math. Z. 233 (2000), [7] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A 35 (2002), R51 R79. [8] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math. 52 (1999), [9] H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney Rivlin rod, Acta Mechanica 127 (1998), [10] D. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. Math. 92 (1970), [11] B. Fuchssteiner and A. Foas, Symplectic structures, their Bäclund transformation and hereditary symmetries, Physica D 4 (1981), [12] R. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), [13] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, Berlin, [14] T. Kato, Quasi-Linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, volume 448, pp , Springer Lecture Notes in Mathematics, 1979.

18 804 A. Constantin and B. Kolev CMH [15] S. Lang, Fundamentals of Differential Geometry, Springer-Verlag, New Yor, [16] J. Milnor, Remars on infinite-dimensional Lie groups, in: Relativity, Groups and Topology (Les Houches, 1983), pp , North-Holland, Amsterdam, [17] G. Misiole, A shallow water equation as a geodesic flow on the Bott Virasoro group, J. Geom. Phys. 24 (1998), Adrian Constantin Department of Mathematics Lund University Box Lund Sweden adrian.constantin@math.lu.se Boris Kolev C. M. I. Université deprovence 39 rue F. Joliot-Curie Marseille Cedex France olev@cmi.univ-mrs.fr (Received: March 28, 2002)