## Abstract

Many important equations of mathematical physics arise geometrically as geodesic equations on Lie groups. In this paper, we study an example of a geodesic equation, the two-component Hunter–Saxton (2HS) system, which displays a number of unique geometric features. We show that 2HS describes the geodesic flow on a manifold, which is isometric to a subset of a sphere. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulae for the solutions of 2HS. We also show that when restricted to functions of zero mean, 2HS reduces to the geodesic equation on an infinite-dimensional manifold, which admits a Kähler structure. We demonstrate that this manifold is in fact isometric to a subset of complex projective space, and that the above constructions provide an example of an infinite-dimensional Hopf fibration.

## 1. Introduction

Several well-known equations of mathematical physics arise geometrically as geodesic equations on Lie groups. The classical example is the motion of a rigid body rotating around its centre of gravity: the motion is described by the classical Euler equation, which is the geodesic equation on *SO*(*n*) endowed with a left-invariant metric defined by the kinetic energy of the body. Another fundamental example is the Euler equation of ideal hydrodynamics: the particles of a fluid moving in a compact *n*-dimensional Riemannian manifold *M* trace out a geodesic curve in the Lie group of volume-preserving diffeomorphisms of *M* equipped with a right-invariant metric defined by the kinetic energy of the fluid [1,2]. Several other physically relevant equations admit similar geometric formulations [3–5]. Once it has been established that an equation admits a formulation of this type, it is tempting to use geometric intuition in order to better understand the behaviour of its solutions; for example, directions of positive or negative curvature are expected to correspond to the existence of stable or unstable perturbations of the motion, respectively.

Arnold demonstrated that the curvature of the group of volume-preserving diffeomorphisms associated with hydrodynamics is negative in some directions while it is positive in others [1]. For most other geodesic equations, such as the Korteweg–de Vries equation and Camassa–Holm equations, similar results apply—the curvature is sometimes negative and sometimes positive [6]. (The geometric interpretation of the Camassa–Holm equation is still useful in establishing criteria for global existence and blow-up [7,8].) On the other hand, there are a few notable exceptions for which the curvature is of a definite sign. In particular, the groups associated with the Hunter–Saxton (HS) equation, the integrable equation proposed in [9], as well as the two-component versions of both of these equations all have positive and constant curvature [9–12]. For the first two of these equations, this property has been ‘explained’ as being a consequence of the fact that the underlying spaces are isometric to subsets of *L*^{2}-spheres [9,11]. Thus, the equations are just the equations for geodesic flow on a sphere in disguise. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulae for the solutions of the equations. However, for the two-component versions of these equations, a similar geometric interpretation has, until now, been lacking. The purpose of the present paper is to ‘explain’ the constant curvature of the two-component Hunter–Saxton (2HS) equation by showing that the underlying space is isometric to (part of) the unit sphere in . Thus, the geometric picture valid for the HS equation extends also to its two-component version. In the case of HS, the sphere is the unit sphere in , whereas in the case of 2HS, it is the unit sphere in . Since the second component of 2HS is related to the complex phase of the functions in , the geometric picture associated with 2HS reduces to that of HS when the second component vanishes. Even though we restrict our attention to 2HS in this paper, we expect that the two-component equation proposed in [12] admits a similar geometric interpretation.

We will also consider the restriction of 2HS to solutions (*u*,*ρ*), where *ρ* has zero mean. After showing that the underlying space in this case is an infinite-dimensional Kähler manifold with positive (but non-constant) curvature, we will show that it is in fact isometric to a subset of complex projective space.

Geometrically, the above constructions provide an example of an infinite-dimensional Hopf fibration. Indeed, the circle *S*^{1} acts on functions in the unit sphere by multiplication by a constant phase. Since the quotient manifold is the infinite-dimensional complex projective space , it follows that fibres over as
1.1The 2HS equation is (up to isometry) the geodesic equation on , while the restriction of 2HS to functions *ρ* of zero mean is the geodesic equation on ; the two equations are related by the Hopf fibration (1.1).

Let us finally point out that geodesic flows on spheres and Hopf fibrations also arise in the analysis of the classical Kepler problem. (Recall that the Kepler problem consists of determining the motion of two point masses interacting under an inverse square force law.) It was noted by Moser that the flow arising in the Kepler problem restricted to the manifold of constant energy *E*<0 is equivalent (up to a rescaling of time) to the geodesic flow on a sphere. Moreover, the trajectory space of the covering flow on the universal covering space *S*^{3} is a two-dimensional sphere *S*^{2}, and the corresponding map
is the classical Hopf fibration [13,14].

After recalling some preliminaries in §2, we establish that the space associated with 2HS is isometric to part of a sphere in §3. In §4, we analyse solutions of the initial-value problem for 2HS. In §5, global properties of the geodesic flow are investigated. In §6, we consider the restriction of 2HS to solutions (*u*,*ρ*) where *ρ* has zero mean, and show that it describes geodesic flow on a Kähler manifold. In §7, we show that this Kähler manifold is isometric to a subset of complex projective space and that the above constructions provide an example of a Hopf fibration. Section 8 contains some concluding remarks.

## 2. Preliminaries

*The HS and 2HS equations*. Let *S*^{1} denote the circle of length one. The periodic HS equation
2.1arises in the study of nematic liquid crystals, with *u*(*t*,*x*) being a real-valued function of a space variable *x* and a slow time variable *t* [15]. Geometrically, HS is the equation for geodesic flow on the Lie group Diff_{0}(*S*^{1}) of diffeomorphisms of the circle *S*^{1} with a designated fixed point [16], endowed with the right-invariant metric given at the identity by
The space Diff_{0}(*S*^{1}) equipped with the -metric is isometric to a subset of the unit sphere in [11].

The 2HS system
2.2where *u*(*t*,*x*) and *ρ*(*t*,*x*) are real-valued functions, is a natural generalization of (2.1). Just like the HS equation, 2HS is an integrable system with an associated Lax pair formulation and a bi-Hamiltonian structure, see [17,18]. Geometrically, (2.2) is the equation for geodesic flow on the semidirect product Lie group , where denotes the circle of length 4*π*, denotes the space of (sufficiently smooth) maps , and the group *G* is endowed with the right-invariant metric given at the identity by
2.3

*Diffeomorphism groups*. In order to set the stage for the rigorous study of (2.2) as a geodesic equation, we need to introduce some notation. Let . Let Diff^{s}(*S*^{1}) denote the Banach manifold of orientation-preserving diffeomorphisms of *S*^{1} of Sobolev class *H*^{s}. We let denote the subgroup of Diff^{s}(*S*^{1}) consisting of diffeomorphisms *φ* that keep the point 0∈*S*^{1}≃[0,1) fixed, i.e.
Let and denote the Hilbert spaces of real-valued and complex-valued functions on *S*^{1} of Sobolev class *H*^{s}, respectively. Using the identification
2.4we can view as an open subset of the closed hyperplane , where is the closed linear subspace
Thus, (2.4) provides a global chart for the Banach manifold .

Let consist of all maps of Sobolev class *H*^{s−1}. Let *G*^{s} denote the semidirect product with multiplication given by
where ° denotes composition and the addition in the second component is pointwise addition of angles, i.e. the addition takes place in . is a Banach manifold modelled on the space ; it is the disjoint union of a countable number of components distinguished by the winding number of their elements. It follows that *G*^{s} also is a Banach manifold. The neutral element of *G*^{s} is (id,0) and (*φ*,*α*) has the inverse (*φ*^{−1},−*α*°*φ*^{−1}). The metric 〈⋅,⋅〉 on *G*^{s} is defined at the identity by (2.3) and extended to all of *G*^{s} by right invariance, i.e.
2.5where *U*=(*U*_{1},*U*_{2}) and *V* =(*V* _{1},*V* _{2}) are elements of .

## 3. A sphere

We will prove that the weak Riemannian manifold (*G*^{s},〈⋅,⋅〉) is isometric to a subset of the unit sphere in . Let denote the unit sphere in and let denote the elements in that are of Sobolev class *H*^{s}, that is,
is a Banach manifold modelled on the closed subspace of functions orthogonal to the constant function ,
where denotes the component of *h* along the space spanned by 1. Indeed, let
3.1denote the stereographic projection from the ‘south pole’ −1 with inverse
Similarly, define the stereographic projection *σ*_{N} from the ‘north pole’ 1 by
Together, the two charts defined by *σ*_{S} and *σ*_{N} cover and determine its manifold structure.

Let denote the open subset of of nowhere vanishing functions,
3.2We equip with the manifold structure inherited from and the weak Riemannian metric 〈⋅,⋅〉_{L2} inherited from , i.e.
whenever .

### Theorem 3.1

*The space (G*^{s}*,〈⋅,⋅〉) is isometric to a subset of the unit sphere in* *. More precisely, for any* *the map* *defined by
**is a diffeomorphism and an isometry.*

### Proof.

If , then the function satisfies *φ*(0)=0, *φ*(1)=1, and *φ*_{x}>0, while the function belongs to . Thus, the inverse of *Φ* is given explicitly by
3.3This shows that *Φ* is bijective. Since both *Φ* and *Φ*^{−1} are smooth, *Φ* is a diffeomorphism. Using
we find that
whenever (*U*_{1},*U*_{2}) and (*V* _{1},*V* _{2}) belong to *T*_{(φ,α)}*G*^{s}. This shows that *Φ* is an isometry. □

It follows immediately from theorem 3.1 that the sectional curvature of *G*^{s} is constant and equal to 1. This result was already proved in a different way in [10].

### Corollary 3.2

*The space* (*G*^{s},〈⋅,⋅〉) *has constant sectional curvature equal to* 1.

### Proof.

In view of theorem 3.1, it is enough to prove that the unit sphere in has constant sectional curvature equal to 1. As in the finite-dimensional case, this can be proved using the Gauss–Codazzi formula. Indeed, letting *n* denote the outward normal to the sphere, the second fundamental form *Π* is given by
where *X*,*Y* are vector fields on . Consequently, if *X* and *Y* are orthonormal, the curvature tensor *R* on the unit sphere satisfies
□

By pulling back the covariant derivative on the sphere , we can determine the metric connection on *G*^{s}. Let . Then, *A* is an isomorphism,
Let *A*^{−1} be its inverse given by
whenever .

### Corollary 3.3

*The metric covariant derivative on G*^{s} *is given by*
3.4*where the Christoffel map Γ is defined for u*=(*u*_{1},*u*_{2}), *v*=(*v*_{1},*v*_{2})∈*T*_{(id,0)}*G*^{s} *by*
3.5a*and extended to all of G*^{s} *by right invariance*,
3.5b

### Proof.

Let ∇′ be the metric connection on . The metric covariant derivative ∇ on *G*^{s} is the pull-back of ∇′ by *Φ*, i.e.
Right invariance implies that it is enough to verify (3.4) at the identity (id,0). We have
and
where *Z*↦*Z*^{t}=*Z*−〈*Z*,*f*〉_{L2}*f* is the orthogonal projection of *Z* onto . Thus,
3.6Let *u*=(*u*_{1},*u*_{2}) and *v*=(*v*_{1},*v*_{2}) be the values of the vector fields *X* and *Y* at the identity, respectively. Evaluation of (3.6) at *f*=1 yields
It follows that
which proves (3.4). □

### Remark 3.4

The Christoffel map (3.5) defines a smooth spray on *G*^{s}, i.e. the map
is smooth (see [19] for a proof in a similar situation).

By definition, the geodesics on *G*^{s} are the solutions (*φ*(*t*),*α*(*t*)) of the equation ∇_{(φt,αt)}(*φ*_{t},*α*_{t})=0, i.e.
3.7Theorem 3.1 immediately leads to explicit formulae for the geodesics in *G*^{s}.

### Corollary 3.5

*Let* . *Let* *be the geodesic in* *G*^{s} *such that* (*φ*(0),*α*(0))=(id,0) *and* (*φ*_{t}(0),*α*_{t}(0))=(*u*_{0},*ρ*_{0})∈*T*_{(id,0)}*G*^{s} *with maximal time of existence* *T*_{s}>0. *Then*, (*φ*(*t*),*α*(*t*)) *is given by*
3.8*that is*,
3.9a*and*
3.9b*where the speed c*>0 *of the geodesic is given by* *The maximal existence time T*_{s} *is independent of* *in the sense that if* (*u*_{0},*ρ*_{0})∈*T*_{(id,0)}*G*^{r} *with* , *then* *T*_{r}=*T*_{s} *for all* . *Moreover, the geodesic* (*φ*(*t*),*α*(*t*)) *exists globally* (*i.e.* ) *if and only if* *ρ*_{0}(*x*)≠0 *for all* *x*∈*S*^{1}.

### Proof.

The geodesic *f*(*t*) on the sphere starting at the constant function 1 with initial velocity *f*_{t}(0) is the great circle given explicitly by
3.10where *c*=∥*f*_{t}(0)∥_{L2} denotes its speed. Indeed, viewing *f*(*t*) as a curve in , we have *f*_{tt}=−*c*^{2}*f*. Hence, the orthogonal projection of *f*_{tt}(*t*) onto the tangent space vanishes for every *t*. By definition of the induced connection ∇′ on , this shows that ∇′_{ft}*f*_{t}≡0. Equation (3.8) now follows from theorem 3.1 and (3.10). Equation (3.9a) then follows from (3.3).

The geodesic (*φ*(*t*),*α*(*t*)) persists as long as
3.11remains in the domain . The maximal existence time *T*_{s} is therefore determined by the time at which *f*(*t*) hits the boundary of , i.e.
It is clear from this expression that *T*_{s} is independent of .

In order to characterize the globally defined geodesics, we need to show that *f*(*t*,*x*)≠0 for all *x*∈*S*^{1} and *t*≥0 if and only if *ρ*_{0}(*x*)≠0 for all *x*∈*S*^{1}. Fix *x*∈*S*^{1}. Clearly, by (3.11), *f*(*t*,*x*)≠0 for all *t* if *ρ*_{0}(*x*)≠0. Conversely, if *ρ*_{0}(*x*)=0, then,
and for any real number *u*_{0x}(*x*), there always exists a *t*≥0 such that this expression vanishes (take *t*=*π*/2*c* if *u*_{0x}(*x*)=0 and if *u*_{0x}(*x*)≠0). □

## 4. Solutions of the Hunter–Saxton system

The geometric picture developed above yields explicit expressions for the solutions of the 2HS equation (2.2). It turns out that there exist solutions of 2HS that break in finite time as well as solutions that exist globally. More precisely, we will show that a solution with initial data (*u*_{0},*ρ*_{0}) breaks in finite time if and only if *ρ*_{0}(*x*) vanishes at some *x*∈*S*^{1}.

We can write 2HS in the following form suitable for the formulation of weak solutions: 4.1

### Proposition 4.1

*Let* . *Let* (*φ*,*α*):*J*→*G*^{s} *be a C*^{2}-*curve where* *is an open interval and define* (*u*,*ρ*) *by* (2.6). *Then*,
4.2*and* (*φ*,*α*) *is a geodesic on* *J* *if and only if* (*u*,*ρ*) *satisfies the 2HS equation* (4.1) *for* *t*∈*J*.

### Proof.

Equation (4.2) follows since, if , the composition map (*f*,*ψ*)↦*f*°*ψ* is *C*^{r} as a map , while the inversion map *ψ*↦*ψ*^{−1} is *C*^{r} as a map Diff^{q+r}(*S*^{1})→Diff^{q}(*S*^{1}) (cf. [2]).

Using the right invariance of *Γ*, the geodesic equation (3.7) can be rewritten as
which is exactly equation (4.1). □

### Theorem 4.2

*Let* *. Let* *. Then, there exists a unique solution (u,ρ) of the 2HS equation (4.1) such that
*4.3*where T*_{s}*>0 is the maximal existence time. The solution is given by
*4.4*where the curve (φ(t),α(t)) in G*^{s} *is given explicitly in terms of (u*_{0}*,ρ*_{0}*) by (3.9a). The maximal existence time T*_{s} *is independent of* *in the sense that if* *with* *, then T*_{r}*=T*_{s} *for all* *. Moreover, the solution (u(t),ρ(t)) exists globally (i.e.* *) if and only if ρ*_{0}*(x)≠0 for all x∈S*^{1}*.*

### Proof.

It follows from proposition 4.1 that (*u*,*ρ*) as defined in (4.4) is a solution of 2HS satisfying (4.3) that exists at least as long as the geodesic (*φ*,*α*) does. The theorem will follow from corollary 3.5, if we can show that the maximal existence time *T*_{s} of the solution (*u*,*ρ*) in (4.3) is in fact equal to the maximal existence time of (*φ*,*α*). Suppose that (*u*,*ρ*) is a solution of 2HS satisfying (4.3). Then the map
is *C*^{1}. Thus, there exists a unique solution of the ODE
such that (*φ*(0),*α*(0))=(id,0) and (*φ*,*α*)∈*C*^{1}([0,*T*_{s});*G*^{s−2}). Then, *F*(*t*,(*φ*,*α*))∈*C*^{1}([0,*T*_{s});*G*^{s−2}) so that in fact (*φ*,*α*)∈*C*^{2}([0,*T*_{s});*G*^{s−2}). Since (*u*,*ρ*) satisfies 2HS, (*φ*,*α*) is a *C*^{2}-geodesic. But since the spray is smooth, this implies that , i.e. the maximal existence time of (*φ*,*α*) as a geodesic in *G*^{s−2} is at least *T*_{s}. Since the maximal existence time of (*φ*,*α*) is independent of *s*, this shows that the existence times of (*φ*,*α*) in *G*^{s} and of (*u*,*ρ*) in (4.3) coincide. □

### Corollary 4.3

*All solutions of 2HS are periodic in time with period* 2*π*. *If* (*u*,*ρ*) *is a solution with maximal existence time* *T*>0, *then either* *or T*<*π*.

### Proof.

*T* is the smallest time for which the corresponding geodesic in hits the boundary of . Since is invariant under the antipodal map *f*↦−*f* on , it follows that this happens for *t*<*π* or it does not happen at all. □

## 5. Global behaviour of geodesics

The last statement of corollary 3.5 gives a characterization of the geodesics on *G*^{s} starting at the identity that exists for all times. In this section, we will elaborate further on the global properties of geodesics on *G*^{s}.

We begin by describing the geodesic flow on the sphere . We let denote the (Riemannian) exponential map on restricted to the tangent space at the constant function 1. The next lemma expresses the fact that given any point , there exists a unique great circle passing through 1 and *f*, unless *f*=±1, in which case, there exists an infinite number of such great circles. Thus, the geodesic flow on behaves as can be expected by analogy with the finite-dimensional case.

### Lemma 5.1

*The exponential map* *on the sphere* *satisfies*
*where the unit length vector* *and the real number* *r*_{0}∈(0,*π*) *are given by*
5.1

### Proof.

Let . If has length 1, we have (cf. equation (3.10))
Thus, if and only if
5.2Applying 〈⋅,1〉_{L2} to both sides of this equation, we find
and the lemma now follows from (5.2). □

Given two points such that *f*≠±*g*, lemma 5.1 implies that there is a unique geodesic of length *r*_{0}∈(0,*π*) joining *f* to *g*; we call this the *short geodesic segment from f to g*. There is also a unique geodesic of length 2

*π*−

*r*

_{0}∈(

*π*,2

*π*) joining

*f*to

*g*, which goes around the sphere in the opposite direction; we call this the

*long geodesic segment from*. If

*f*to*g**f*,

*g*belong to , we may ask whether the short and long geodesic segments connecting

*f*to

*g*are also contained in . Clearly, since is invariant under the antipodal map

*f*↦−

*f*, the short geodesic segment lies in whenever the long segment does.

### Proposition 5.2

*Let* *and suppose that f*≠±*g*. *The short geodesic segment from f to g is contained in* *if and only if* *for all* *x*∈*S*^{1}. *The long geodesic segment from* *f* *to* *g* *is contained in* *if and only if* *for all* *x*∈*S*^{1}.

### Proof.

Let *Φ*((*φ*,*α*))=*f* and *Φ*((*ψ*,*β*))=*g* be two points in . Right invariance implies that there exists a geodesic from (*φ*,*α*) to (*ψ*,*β*) in *G*^{s+1} if and only if there exists one from (id,0) to (*φ*,*α*)(*ψ*,*β*)^{−1}=(*φ*°*ψ*^{−1},(*α*−*β*)°*ψ*^{−1}). Moreover, right translation preserves the length of a geodesic. Hence, the short (long) geodesic segment from *f* to *g* is contained in if and only if the short (long) geodesic segment from *Φ*((id,0))=1 to
is contained in . It is therefore enough to prove the proposition in the case when *g*=1.

Let with *f*≠±1. Let *X*_{0} and *r*_{0} be as in (5.1). We claim that
5.3belongs to for *r*∈(0,*r*_{0}) if and only if for *x*∈*S*^{1}. Indeed, let us fix *x*∈*S*^{1}. Then, for some *r*∈(0,*r*_{0}) if and only if for some *r*∈(0,*r*_{0}). Since the right-hand side of this equation maps the interval (0,*r*_{0}) to , we see that for some *r*∈(0,*r*_{0}) if and only if *f*(*x*)<0. This proves the first half of the proposition. In order to prove the second half, we need to show that the geodesic lies in for if and only if for *x*∈*S*^{1}. This can either be deduced from (5.3) or be seen as a consequence of the last statement of corollary 3.5 using . □

Note that the antipodal involution *f*↦−*f* on corresponds under the isometry *Φ* to the involution (*φ*,*α*)↦(*φ*,*α*+2*π*) of *G*^{s}. Thus, if we use the isometry *Φ* to transfer the result of proposition 5.2 to *G*^{s}, we immediately find the following result.

### Corollary 5.3

*Let* (*φ*,*α*),(*ψ*,*β*)∈*G*^{s} *be distinct points in* *G*^{s} *and suppose that* (*φ*,*α*)≠(*ψ*,*β*+2*π*). *There exists a geodesic joining* (*φ*,*α*) to (*ψ*,*β*) *if and only if e*^{i(α(x)−β(x))/2}≠−1 *for all x*∈*S*^{1}. *This geodesic is unique and has length less than π provided that there exists an x such that e*^{i(α(x)−β(x))/2}=1. *If no such x exists* (*so that e*^{i(α(x)−β(x))/2}≠±1 *for all x*∈*S*^{1}), *then the geodesic is defined on all of* , *is periodic with period* 2*π with respect to an arc-length parameter, and is unique up to the choice of its direction. On the other hand, for any* (*φ*,*α*)∈*G*^{s}, *there exists an infinite number of geodesics joining* (*φ*,*α*) to (*φ*,*α*+2*π*). *All of these geodesics exist globally and are* 2*π*-*periodic with respect to an arc-length parameter*.

Let denote the (Riemannian) exponential map on *G*^{s} restricted to *T*_{(id,0)}*G*^{s}. Using the above results, it is possible to express and its multi-valued inverse explicitly. The following proposition gives the expression for in the case that (*φ*,*α*)≠(id,0) and (*φ*,*α*)≠(id,2*π*).

### Proposition 5.4

*Let* (*φ*,*α*)∈*G*^{s} *and suppose that* (*φ*,*α*)≠(id,0) *and* (*φ*,*α*)≠(id,2*π*). *Then*, *is the empty set if* *e*^{iα(x)/2}=−1 *for some* *x*∈*S*^{1}. *Assuming that* *e*^{iα(x)/2}≠−1 *for all* *x*∈*S*^{1}, *we have*
*where the unit length vector* (*u*_{0},*ρ*_{0})∈*T*_{(id,0)}*G*^{s} *and the real number* *r*_{0}∈(0,*π*) *are given by*
5.4

### Proof.

The expressions in (5.4) follow from lemma 5.1 since *T*_{(id,0)}*Φ*(*u*_{0},*ρ*_{0})=*X*_{0}, where *X*_{0} is as in (5.1) with . The rest follows from corollary 5.3. □

### Remark 5.5

If *α*≡0, the above results reduce to those derived in [11] for HS. Nevertheless, there are big differences between the geometries associated with 2HS and HS. For example, for HS, any two points of the underlying space can be joined by a unique length-minimizing geodesic [11]. By contrast, for 2HS, we have seen that there are points that can be joined by more than one geodesic, as well as points that cannot be joined by any geodesic, even though they lie in the same component of *G*^{s}.

## 6. A Kähler manifold

The mean value of the second component *ρ* of a solution (*u*,*ρ*) of 2HS is conserved, i.e.
Thus, if *ρ* has zero mean initially, it will have zero mean at all later times. This suggests that we consider the following variation of 2HS:
6.1where denotes the orthogonal projection onto the subspace of functions of zero mean.

For solutions such that , equation (6.1) coincides with 2HS. However, we will see that (6.1) possesses some interesting geometric properties not shared by the 2HS equation (2.2). In particular, (6.1) is the geodesic equation on a manifold *K* that admits a Kähler structure.

*The Kähler manifold* *K*^{s}. Let *s*>5/2. Let denote the space with two elements being identified if and only if they differ by a constant phase; the equivalence class of will be denoted by . We define *K*^{s} as the semidirect product with multiplication given by
Let denote the space with two functions being identified if and only if they differ by a constant. Since the constant functions form a closed linear subspace of , the quotient space is a Banach space. The space is a Banach manifold modelled on . Together with the global chart (2.4) for , this turns *K*^{s} into a Banach manifold.

We equip *K*^{s} with the right-invariant metric given at the identity by
6.2Extending the projection *π* to any tangent space by right invariance so that whenever , we have
6.3

We define a connection ∇ on *K*^{s} by
where the Christoffel map *Γ* is defined for *u*=(*u*_{1},[*u*_{2}]), *v*=(*v*_{1},[*v*_{2}]) in *T*_{(id,[0])}*K*^{s} by
6.4aand extended to the tangent space at (*φ*,[*α*])∈*K*^{s} by right invariance
6.4bWe also define a (1,1)-tensor *J* and a two-form *ω* on *K*^{s} by
6.5and
6.6whenever (*U*_{1},[*U*_{2}]),(*V* _{1},[*V* _{2}])∈*T*_{(φ,[α])}*K*^{s}. Note that *ω* and *J* are right invariant. Indeed, a change of variables in (6.6) shows that
while right invariance of *J* follows by a simple calculation,
if (*U*_{1},[*U*_{2}])=(*u*_{1},[*u*_{2}])°*φ*∈*T*_{(φ,[α])}*K*^{s} and *R*_{(φ,[α])} denotes right multiplication by (*φ*,[*α*]). We refer to [20] for an introduction to differential forms and tensor fields on Banach manifolds.

### Theorem 6.1

*K*^{s} *is a Kähler manifold. In fact, letting g denote the metric 〈⋅,⋅〉 on K*^{s}, *s*>5/2, *the following hold*:

(a)

*g is a smooth metric on K*^{s}*and*∇*is a smooth connection compatible with g*;(b)

*ω is a symplectic form on K*^{s}*compatible with ∇, i.e. ω is a smooth non-degenerate closed two-form on K*^{s}*such that ∇ω=0;*(c)

*J is a complex structure on K*^{s}*compatible with ∇, i.e. J is a smooth (1,1)-tensor on K*^{s}*such that J*^{2}*=−I and ∇J=0;*(d)

*the symplectic form ω, the metric g and the complex structure J are compatible, i.e. ω(U,V)=g(JU,V);*(e)

*the metric g is almost Hermitian, i.e. g(U,V)=g(JU,JV); and*(f)

*the Nijenhuis tensor N*^{J}*defined for vector fields X,Y by**vanishes identically.*

### Proof.

Throughout the proof, *u*=(*u*_{1},[*u*_{2}]), *v*=(*v*_{1},[*v*_{2}]) and *w*=(*w*_{1},[*w*_{2}]) will denote elements of *T*_{(id,[0])}*K*^{s}.

*Proof of* (*a*). Smoothness of *g* follows since (6.3) depends smoothly on *φ* and *α* as a bilinear map from to . ∇ defines a smooth connection because, as in the case of *G*^{s} above, the Christoffel map *Γ* defined in (6.4) defines a smooth spray on *K*^{s}, i.e. the map
is smooth (cf. remark 3.4).

In order to show that ∇ and *g* are compatible, we need to show that
for any vector fields *X*,*Y*,*Z* on *K*^{s}. By right invariance, it is enough to verify this identity at the identity (id,[0]). Moreover, using the argument of Ebin & Marsden [2], pp. 129–130, it is enough to prove it when *X*,*Y*,*Z* are right-invariant vector fields. Thus, assume that
6.7Then, the function 〈*Y*,*Z*〉 is constant so that *X*〈−*Y*,*Z*〉=0. Moreover,
Simplification using integration by parts leads to

*Proof of* (*b*). First note that *ω* is a smooth two-form on *K*^{s} because the right-hand side of (6.6) is skew-symmetric in *U*,*V* and independent of (*φ*,[*α*]). Moreover, *ω* is non-degenerate because if *ω*((*U*_{1},[*U*_{2}]),(*V* _{1},[*V* _{2}]))=0 for all *V* _{1},*V* _{2}, then *U*_{1}=*U*_{2x}=0, which means that (*U*_{1},[*U*_{2}])=0. Since the local expression (6.6) of *ω* is independent of (*φ*,[*α*]), it follows that *dω*=0.

It remains to prove that ∇*ω*=0. We need to verify that
6.8for all vector fields *X*,*Y*,*Z*. As in the proof of the identity ∇*g*=0, it is enough to verify (6.8) at the identity and in the case that *X*,*Y*,*Z* are right invariant. Thus, let *X*,*Y*,*Z* be right-invariant vector fields as in (6.7). Then, *Xω*(*Y*,*Z*)=0 and
In view of the identity we can write the right-hand side as
Since the part of this expression that is antisymmetric in *v* and *w* vanishes, we deduce that *ω*(∇_{X}*Y*,*Z*)+*ω*(*Y*,∇_{X}*Z*)=0.

*Proof of* (*c*). Smoothness of *J* follows since the right-hand side of (6.5) depends smoothly on *φ* as a linear map from to itself. Since
we have *J*^{2}(*U*_{1},[*U*_{2}])=−(*U*_{1},[*U*_{2}]), showing that *J* is an almost complex structure. It only remains to prove that ∇*J*=0. This can either be seen as a consequence of ∇*g*=∇*ω*=0 and the statement (*d*) proved below, or be established directly as follows. The covariant derivative of the (1,1)-tensor *J* is given locally by
We compute each of the terms in turn,
Since evaluation at the identity yields
where (*u*_{1},[*u*_{2}]) and (*v*_{1},[*v*_{2}]) are the values of *X* and *Y* at the identity, respectively. Moreover,
and
The sum of the preceding three equations vanishes; thus, ∇*J*=0.

*Proof of* (*d*). We have

*Proof of* (*e*). This is a simple calculation,

*Proof of* ( *f*). It is enough to verify this at the identity by right invariance. We let [⋅,⋅] be the Lie bracket on *T*_{(id,[0])}*K*^{s} induced by right-invariant vector fields. Then,
6.9and the sum of these four equations vanishes after simplification. Thus, *N*^{J}=0. □

Writing (6.1) in the weak form 6.10we find the following result.

### Proposition 6.2

*Let* . *Let* (*φ*,[*α*]):*J*→*K*^{s} *be a* *C*^{2}-*curve where* *is an open interval and define* (*u*,*ρ*) *by* (2.6). *Then*,
6.11*and* (*φ*,[*α*]) *is a geodesic if and only if* (*u*,[*ρ*]) *satisfies* (6.10).

## 7. Complex projective space and the Hopf fibration

In this section, we will show that the Kähler manifold *K*^{s} introduced in §6 is isometric to a subset of complex projective space. Under this isometry, the metric 〈⋅,⋅〉 on *K*^{s} is simply the Fubini–Study metric, and the Kähler structure on *K*^{s} corresponds to the canonical Kähler structure on . Moreover, we will show that the fibration
7.1that arises because the circle acts on *G*^{s} by addition of a constant phase in the second component and the quotient space is *K*^{s}, corresponds under the above isometries to the Hopf fibration of over .

We first define the relevant infinite-dimensional complex projective space in detail. Let , where *f*∼*g* if and only if *f*(*x*)=*cg*(*x*) for some . For *s*>5/2, let denote the elements in of Sobolev class *H*^{s}, i.e.
Let be the natural projection. We turn into a Banach manifold modelled on the space
as follows. For each *x*_{0}∈*S*^{1}, we let *W*_{x0} be the open subset of defined by
and let be the map
The collection of charts {(*W*_{x0},*φ*_{x0})}_{x0∈S1} covers . Moreover, for any *x*_{0},*x*_{1}∈*S*^{1}, the transition map given by
is smooth. This defines the manifold structure on .

The circle acts by isometries on the unit sphere by multiplication by a constant phase,
The quotient space is and the restriction of *q* to is the quotient map for this action. The action determines, for each , an orthogonal splitting of the tangent space according to
where the vertical and horizontal subspaces are given by
7.2and
respectively. The Fubini–Study metric on is defined by
where *X*^{h} and *Y* ^{h} denote the horizontal components of *X*,*Y* . The orthogonal projection of onto is given by

Let be the image under *q* of the subset of nowhere vanishing functions defined in (3.2). Let *p*:*G*^{s}→*K*^{s} denote the projection
7.3Recall that a smooth submersion *F* from *M* to *N*, where *M* and *N* are (possibly weak) Riemannian manifolds, is a *Riemannian submersion* if the restriction of *T*_{p}*F* to the horizontal subspace is an isometry onto *T*_{p}*N* for each *p*∈*M*.

### Theorem 7.1

*Let s>5/2. The map* *defined by
**is a diffeomorphism and an isometry. Moreover, the natural projections p:G*^{s}*→K*^{s} *and* *are Riemannian submersions and the following diagram commutes:
*7.4*where Φ denotes the isometry of theorem 3.1.*

### Proof.

The map *Ψ* is bijective with inverse given by
Since *Ψ* and *Ψ*^{−1} are smooth, *Ψ* is a diffeomorphism. The commutativity of the diagram (7.4) follows by construction.

We next verify that the projection is a Riemannian submersion. Smoothness of *q* can be verified in local charts. For example, the local representative of *q* with respect to the charts determined by *φ*_{x0} and the stereographic projection *σ*_{S} (cf. (3.1)) is the smooth map
By definition of the Fubini–Study metric, *q* is a Riemannian submersion.

The projection *p*:*G*^{s}→*K*^{s} is also a Riemannian submersion. Indeed, smoothness of *p* is immediate, and for each (*φ*,*α*)∈*G*^{s}, *p* determines the splitting
where the vertical and horizontal subspaces are defined by
and
respectively. The orthogonal projections onto the vertical and horizontal subspaces are given by
and
7.5respectively. Let *U*^{h}=(*U*_{1},*U*_{2}) and *V* ^{h}=(*V* _{1},*V* _{2}) be horizontal vectors in *T*_{(φ,α)}*G*^{s}. Then, since *Tp*(*U*_{1},*U*_{2})=(*U*_{1},[*U*_{2}]),
showing that *p* is a Riemannian submersion.

Since both *q* and *p* are Riemannian submersions and *Φ* is an isometry, it follows from the commuting diagram (7.4) that *Ψ* also is an isometry. □

### Corollary 7.2

*Let* *e*=(id,[0]) *denote the identity element in* *K*^{s}. *The curvature tensor* *R* *on* *K*^{s} *satisfies*
7.6*where* *u*=(*u*_{1},[*u*_{2}]) *and* *v*=(*v*_{1},[*v*_{2}]) *are elements in* *T*_{e}*K*^{s}. *In particular, the sectional curvature*
*satisfies*
7.7*and* *sec*(*u*,*v*)=4 *if and only if* *Ju* *is a multiple of* *v*.

### Proof.

According to theorem 7.1, *p*:*G*^{s}→*K*^{s} is a Riemannian submersion. The O’Neill formula for Banach manifolds [20] implies that
where *X*^{h}, *Y* ^{h} denote the horizontal lifts of two orthonormal vector fields *X*,*Y* on *K*^{s} and *R*_{Gs} denotes the curvature tensor on *G*^{s}. In view of corollary 3.2 and equation (7.5), this yields
Since
we find (7.6). To prove (7.7), we assume that *u* and *v* are orthogonal. Then,
with equality if and only if *Ju* is a multiple of *v*. □

## 8. Conclusions and remarks

As was noted in §1, the flow of the classical Kepler problem in celestial mechanics is equivalent to the geodesic flow on a sphere. In this paper, we showed that the flow of the 2HS equation (2.2) is also equivalent to the geodesic flow on a sphere, namely, to the geodesic flow on (a subset of) the unit sphere in . Using this geometric picture, we were able to integrate equation (2.2) explicitly. Moreover, an infinite-dimensional example of a Hopf fibration was obtained by considering the restriction of (2.2) to solutions (*u*,*ρ*), where *ρ* has zero mean. The restricted equation describes the geodesic flow on an infinite-dimensional complex projective space with a natural Kähler structure.

Equations (2.1) and (2.2) are the special cases when *M*=*S*^{1} of two more general equations introduced in [9] and [12], which are defined for any compact Riemannian manifold *M*. The first of these equations describes the geodesic flow on the unit sphere in [9], whereas no such sphere interpretation is as yet known for the second equation (although it is known that the underlying space has constant positive curvature [12]). The considerations of this paper suggest that the flow of the equation in [12] is equivalent to the geodesic flow on (a subset of) the unit sphere in . The details of this construction will be considered elsewhere.

We emphasize that the fact that the underlying spaces for the above four geodesic equations (the eqn in [12] and its three special cases (2.1), (2.2) and the eqn in [9]) have constant positive curvature is exceptional in the context of PDEs that arise as geodesic equations. In most cases, the curvature takes on both signs (one exception is the inviscid Burgers equation in one space dimension, for which the curvature is everywhere non-negative; in higher dimensions, this is no longer true).

## Acknowledgements

The author acknowledges support from the EPSRC, UK.

- Received December 11, 2012.
- Accepted March 11, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.