Geometry of diffeomorphism groups, complete integrability and optimal transport
Abstract.
We study the geometry of the space of densities , which is the quotient space of the diffeomorphism group of a compact manifold by the subgroup of volumepreserving diffemorphisms, endowed with a rightinvariant homogeneous Sobolev metric. We construct an explicit isometry from this space to (a subset of) an infinitedimensional sphere and show that the associated EulerArnold equation is a completely integrable system in any space dimension. We also prove that its smooth solutions break down in finite time.
Furthermore, we show that the metric induces the FisherRao (information) metric on the space of probability distributions, and thus its Riemannian distance is the spherical version of Hellinger distance. We compare it to the Wasserstein distance in optimal transport which is induced by an metric on . The geometry we introduce in this paper can be seen as an infinitedimensional version of the geometric theory of statistical manifolds.
March 2, 2018
AMS Subject Classification (2000): 53C21, 58D05, 58D17.
Keywords: diffeomorphism groups, Riemannian metrics, geodesics, curvature, EulerArnold equations, optimal transport, Hellinger distance, integrable systems.
Contents:
 1 Introduction
 2 Geometric background
 3 The spherical geometry of the space of densities
 4 Probability and infinitedimensional geometric statistics
 5 The geodesic equation: solutions and integrability
 6 Global properties of solutions
 7 The general EulerArnold equation of the  metric
 8 The space of metrics and the diffeomorphism group
1. Introduction
The geometric approach to hydrodynamics pioneered by V. Arnold [3] is based on the observation that the particles of a fluid moving in a compact dimensional Riemannian manifold trace out a geodesic curve in the infinitedimensional group of volumepreserving diffeomorphisms (volumorphisms) of . Arnold’s framework is very general. It includes a variety of nonlinear partial differential equations of mathematical physics—in abstract form often referred to as EulerArnold or EulerPoincaré equations.
With a few exceptions, papers on infinitedimensional Riemannian geometry,
including diffeomorphism groups,
tended to focus exclusively on either strong metrics
or weak metrics of type.
On the other hand, in recent years there have appeared a number of interesting nonlinear evolution equations described as geodesic equations on diffeomorphism groups with respect to weak Riemannian metrics of Sobolev type, see e.g., [4, 18, 20] and their references for examples such as the CamassaHolm, the HunterSaxton or the Eulerequations.
In this paper we focus on the metrics both from a differentialgeometric and a dynamical systems perspective. We will show that they arise naturally on (generic) orbits of diffeomorphism groups in the space of all Riemannian metrics on . The main results of this paper concern the geometry of a subclass of such metrics, namely, degenerate rightinvariant Riemannian metrics on the full diffeomorphism group and the properties of solutions of the associated geodesic equations. The normalized metric is given at the identity diffeomorphism by
(\mbox{1.1}) 
It descends to a nondegenerate Riemannian metric on the homogeneous space of right cosets (densities) . Furthermore, it turns out that the corresponding geometry is spherical for any compact manifold . More precisely, we prove that equipped with (\mbox{1.1}) the space is isometric to (a subset of) an infinitedimensional sphere in a Hilbert space. In fact, the metric we define on can be viewed as an analog of the metric introduced by Otto [30] in the theory of mass transport; furthermore the Riemannian distance of (\mbox{1.1}), which will be shown to coincide with the (spherical) Hellinger distance wellknown in probability and mathematical statistics, can be viewed as an analogue of the Wasserstein distance. Remarkably, it also turns out that our metric induces the socalled FisherRao (information) metric and related ChentsovAmari connections which have diverse applications in asymptotic statistics, information theory and quantum mechanics, see e.g., [1, 9].
We derive the EulerArnold equations associated to the general rightinvariant metrics which include as special cases the dimensional (inviscid) Burgers equation, the CamassaHolm equation, as well as variants of the Euler equation. In the particular case of the homogeneous metric (\mbox{1.1}) the EulerArnold equation has the form
(\mbox{1.2}) 
where is a timedependent vector field on with . This equation is a natural generalization of the completely integrable onedimensional HunterSaxton equation [19] which is also known to yield geodesics on the homogeneous space (the quotient of the diffeomorphism group of the circle by the subgroup of rotations), see [20].
We prove that the solutions of (\mbox{1.2}) describe the great circles
on a sphere in a Hilbert space and in particular the equation is
a completely integrable PDE for any number of space variables.
The corresponding complete family of conserved integrals will be constructed
in terms of angular momenta.
We point out that with a few exceptions (such as
the twodimensional KadomtsevPetviashvili, Ishimori, and DaveyStewartson equations)
all known integrable evolution equations are limited to one space dimension.
Furthermore, we show that the maximum existence time for smooth solutions of (\mbox{1.2}) is necessarily finite for any initial conditions, with the norm of the solution growing without bound as approaches the critical time. On the other hand, the geometry of the problem points to a method of constructing global weak solutions of (\mbox{1.2}). We will describe a strategy showing how this can be done using a technique of J. Moser.
It is also of interest to consider the general form of the rightinvariant () Sobolev metric on given at the identity by
(\mbox{1.3}) 
where are vector fields on ,
is the Riemannian volume form,
Finally, a comment on the functional analytic framework we chose for the paper. While our motivations and objectives are directly related to questions in analysis and PDE, in order to better present our geometric ideas we will—with few exceptions—work primarily with objects (function spaces, diffeomorphism groups, etc.) which consist of smooth functions. However, we emphasize that when these objects are equipped with a suitably strong topology (for example, any Sobolev topology with sufficiently large will do for our purposes) then our constructions are rigorously justified in a routine manner. We will not belabour this point and instead refer the reader to the papers [12] and [28] where such questions are considered in greater detail.
The structure of the paper is as follows. In Section 2 we review the geometric background on EulerArnold equations and describe the space of densities used in optimal transport, as well as reductions of the  and type metrics on , its subgroup , and their quotient.
In Section 3 we introduce the homogeneous metric on the space of densities and study its geometry. Generalizing the results of [23, 24] for the case of the circle we show that for any dimensional manifold the space is isometric to a subset of the sphere in with the induced metric. The corresponding Riemannian distance is shown to be the spherical Hellinger distance.
In Section 4 we describe the relation of the metric to geometric statistics and probability. In particular, we show that on the space it plays the role of the classical FisherRao metric. In the case we then use it to introduce the analogues of dual affine connections generalizing the constructions of Chentsov and Amari.
In Section 5 we study local properties of solutions to the corresponding EulerArnold equation and demonstrate its complete integrability, as a geodesic flow on the sphere. Since for our equation reduces to the HunterSaxton equation we thus obtain an integrable generalization of the latter to any space dimension.
In Section 6 we turn to global properties of solutions. We derive an explicit formula for the Jacobian, prove that solutions necessarily break down in finite time and present an approach to construct global weak solutions.
In Section 7 we derive the EulerArnold equation for the general  metric (\mbox{1.3}) and show that several wellknown PDE of mathematical physics can be obtained as special cases. We also discuss the situations in which some of the coefficients , , or are zero.
Finally, in Section 8 we present a geometric construction which yields rightinvariant metrics of the type (\mbox{1.3}) as induced metrics on the orbits of the diffeomorphism group from the canonical Riemannian structures on the spaces of Riemannian metrics and volume forms on the underlying manifold .
Acknowledgements. We thank Aleksei Bolsinov, Nicola Gigli and Emanuel Milman for helpful suggestions.
2. Geometric background
2.1. The EulerArnold equations
In this section we describe the general setup which is convenient to study geodesics on Lie groups and homogeneous spaces equipped with rightinvariant metrics.
Let be a (possibly infinitedimensional) Lie group with a group operation denoted by . In our main examples group elements will be diffeomorphisms and the operation will be their composition. We shall use to denote the Lie algebra of , where is the identity element. For any the group adjoint is the map given by the differential
where stands for a lefttranslation on the group while is the corresponding righttranslation . The algebra adjoint is given by
where is any curve in with and . If the group operation is composition of diffeomorphisms then in terms of the standard Lie bracket of vector fields we have .
We equip with a rightinvariant (possibly weak) Riemannian metric which is determined by an inner product on the tangent space at the identity
where and . The EulerArnold equation on the Lie algebra for the corresponding geodesic flow has the form
(\mbox{2.1}) 
with and where the bilinear operator on is defined by
(\mbox{2.2}) 
Equation (\mbox{2.1}) describes the evolution in the Lie algebra of the vector obtained by righttranslating the velocity along the geodesic in starting at the identity with initial velocity . The geodesic itself can be obtained by solving the Cauchy problem for the flow equation
Remark 2.1.
Rewriting Equation (\mbox{2.1}) on the dual space in the form
gives a conservation law expressing the fact that is confined to one and the same coadjoint orbit during the evolution.
Remark 2.2.
Let be a closed subgroup of . A rightinvariant metric on descends to an invariant (under the right action of ) metric on the homogeneous space if and only if the projection of the metric to , the orthogonal complement to in the group , is biinvariant with respect to the subgroup action. (If the metric in is degenerate along the subgroup then this condition reduces to the metric biinvariance with respect to the action, see e.g., [20]. We shall consider the general case in Section 2.4 below.) The corresponding EulerArnold equation is then defined similarly as long as the metric is nondegenerate on the quotient .
2.2. Examples: equations of fluid mechanics
We list several equations of mathematical physics that arise as geodesic flows on diffeomorphism groups.
Let be the group of volumepreserving diffeomorphisms (volumorphisms) of a closed Riemannian manifold . Consider the rightinvariant metric on generated by the inner product
(\mbox{2.3}) 
In this case the EulerArnold equation (\mbox{2.1}) is the Euler equation of an ideal incompressible fluid in
(\mbox{2.4}) 
where is the velocity field and is the pressure function, see [3]. In the vorticity formulation the 3D Euler equation becomes
Consider the rightinvariant metric on given by the inner product
The corresponding EulerArnold equation is sometimes called the Euler– (or Lagrangianaveraged) equation and in 3D has the form
(\mbox{2.5}) 
see e.g. [18].
Another source of examples is related to various rightinvariant Sobolev metrics on the group of all circle diffeomorphisms, as well as its onedimensional central extension, the Virasoro group. Of particular interest are those metrics whose EulerArnold equations turn out to be completely integrable.
On with the metric defined by the product the EulerArnold equation (\mbox{2.1}) becomes the (rescaled) inviscid Burgers equation
(\mbox{2.6}) 
while the product yields the CamassaHolm equation
(\mbox{2.7}) 
Similarly, the homogeneous part of the product gives rise to the HunterSaxton equation
(\mbox{2.8}) 
More precisely, in the latter case one considers the quotient whose tangent space at the identity coset , i.e., the coset corresponding to the identity diffeomorphism, can be identified with periodic functions of zero mean. The rightinvariant metric on is defined by the inner product on such functions
and the corresponding EulerArnold equation is given by the HunterSaxton equation (\mbox{2.8}).
We also mention that if is the Virasoro group equipped with the rightinvariant metric then the EulerArnold equation is the periodic Kortewegde Vries equation
which is a shallow water approximation and the classical example of an infinitedimensional integrable system. We refer the reader to [20] for more details on these constructions.
Remark 2.3.
The HunterSaxton equation will be of particular interest to us in this paper. In [23, 24] Lenells constructed an explicit isometry between and a subset of the unit sphere in and described the corresponding solutions of Equation (\mbox{2.8}) in terms of the geodesic flow on the sphere. Although the solutions exist classically only for a finite time they can be extended beyond the blowup time as weak solutions, see [25]. In the sections below, we shall show that this phenomenon can be established for flows on manifolds of arbitrary dimension.
2.3. The optimal transport and Otto’s calculus
Given a volume form on there is a natural fibration of the diffeomorphism group
over the space of volume forms of fixed total volume .
More precisely, the projection onto the quotient space
defines a smooth ILH principal bundle
see Moser [29]. Alternatively, let denote the RadonNikodym derivative of with respect to the reference volume form . Then the base (as the space of constantvolume densities) can be regarded as a convex subset of the space of smooth functions on
In this case the projection map can be written explicitly as where denotes the Jacobian of computed with respect to , that is, .
The fact that implies that
(\mbox{2.9}) 
As a consequence, the projection satisfies whenever , i.e., whenever . Thus is constant on the left cosets and descends to an isomorphism between the quotient space of left cosets to the space of densities.
The group carries a natural metric
(\mbox{2.10}) 
where and . This metric is neither left nor rightinvariant,
although it becomes rightinvariant when restricted to the subgroup of
volumorphisms and becomes leftinvariant only on the subgroup of isometries.
Its significance comes from the fact that a curve in is a geodesic
if and only if
is a geodesic in for each .
The induced metric on the base is then
(\mbox{2.11}) 
where and solve and with meanzero functions and considered as elements of the tangent space at .
The geodesic equation of the metric (\mbox{2.10}) on is
i.e., individual particles of the geodesic flow of diffeomorphisms move along the geodesics in until they cross (and a smooth solution ceases to exist). In Eulerian coordinates, using , the geodesic equation can be rewritten as the pressureless Euler (or, inviscid Burgers) equation
(\mbox{2.12}) 
and the induced geodesic equation on the quotient space reads
(\mbox{2.13}) 
where and
is a smooth function.
The associated Riemannian distance in between two measures and has an elegant interpretation as the cost of transporting one density to the other
(\mbox{2.14}) 
with the infimum taken over all diffeomorphisms such that and where denotes the Riemannian distance on ; see [5] or [30]. The function is the Wasserstein (or KantorovichRubinstein) distance between and and is of fundamental importance in optimal transport theory.
2.4. Homogeneous metrics on
In this section we formulate a condition under which metrics on descend to the homogeneous space of densities and describe several examples. The noninvariant metric used in optimal transport, as well as our main example, the rightinvariant metric (\mbox{1.1}), both descend to the quotient. But other natural candidates, such as the rightinvariant metric on or the full metric, do not.
We start with a general observation about rightinvariant metrics. Let be a closed subgroup of a group .
Proposition 2.5.
A rightinvariant metric on descends to a rightinvariant metric on the homogeneous space if and only if the inner product restricted to (the orthogonal complement of ) is biinvariant with respect to the action by the subgroup , i.e., for any and any one has
(\mbox{2.15}) 
Proof.
The proof repeats with minor changes the proof for the case of a metric that is degenerate along a subgroup ; see [20]. In the latter case condition (\mbox{2.15}) reduces to biinvariance with respect to the action and there is no need to confine to the orthogonal complement . We only observe that in order to descend the orthogonal part of the metric must be invariant
for any . The corresponding condition for the Lie algebra action is obtained by differentiation. ∎
Remark 2.6.
It can be checked that the condition in Proposition 2.5 is precisely what one needs in order for the projection map from to to be a Riemannian submersion, i.e., that the length of every horizontal vector is preserved under the projection. Since the metric on is assumed rightinvariant, this condition reduces to one that can be checked in the tangent space at the identity.
Example 2.7.
The degenerate rightinvariant metric (\mbox{1.1}) on descends to a nondegenerate metric on the quotient . The skew symmetry condition (\mbox{2.15}) in this case will be verified in Section 7 below (Corollary 7.4).
Example 2.8.
The rightinvariant metric (\mbox{2.3}) does not verify (\mbox{2.15}) and hence does not descend to . In fact, if we set then for any vector field with integration by parts gives
where we used the identity . It is not difficult to find and such that the above integral is nonzero. For example, we can take to be the divergencefree part of the field and arrange for a suitable so that .
Similarly, it follows that the full metric on obtained by righttranslating the  product (\mbox{1.3}) also fails to descend to a metric on . Note that the term in (\mbox{1.3}) does not contribute in this case.
Example 2.9.
As already pointed out in Section 2.3 the noninvariant metric (\mbox{2.10}) descends to Otto’s metric on the quotient space whose Riemannian distance is the Wasserstein distance on . This metric is invariant under the action of .
Remark 2.10.
One can also consider noninvariant Sobolev metrics analogous to the noninvariant metric (\mbox{2.10}) on . If the manifold is flat then (identifying a neighbourhood of the identity in the diffeomorphism group with a neighbourhood of zero in a vector space) the energy functional of such a metric evaluated on a curve will have the form
The first term will not be affected by a volumepreserving change of variables. However, the terms involving derivatives in the space variable ( and ) will not be conserved in general. This argument can be developed to show that among metrics of this type the noninvariant metric (corresponding to ) is the only one descending to the homogeneous space of densities .







3. The spherical geometry of the space of densities
In this section we study the homogeneous space of densities on a closed dimensional Riemannian manifold equipped with the rightinvariant metric induced by the inner product (\mbox{1.1}), that is
(\mbox{3.1}) 
for any and . It corresponds to the term in the general () Sobolev metric (\mbox{1.3}) of the Introduction in which, to simplify calculations, we set . (We will return to the case of in Sections 7 and 8.)
The geometry of this metric on the space of densities turns out to be particularly remarkable.
Indeed, we prove below that endowed with the metric (\mbox{3.1})
is isometric to an open subset of a round sphere in the space of squareintegrable functions
on .
3.1. An infinitedimensional sphere
We begin by constructing an isometry between the homogeneous space of densities and an open subset of the sphere of radius
in the Hilbert space .
As before, we let denote the Jacobian of with respect to the reference form and let stand for the total volume of .
Theorem 3.1.
The map given by
defines an isometry from the space of densities equipped with the metric (\mbox{3.1}) to an open subset of the sphere of radius
with the standard metric.
For the map is a diffeomorphism between and the convex open subset of which consists of strictly positive functions on .
Proof.
First, observe that the Jacobian of any orientationpreserving diffeomorphism is a strictly positive function. Next, using the change of variables formula, we find that
which shows that maps diffeomorphisms into . Furthermore, observe that since for any we have
it follows that is welldefined as a map from .
Next, suppose that for some diffeomorphisms and we have . Then from which we deduce that is injective. Moreover, differentiating the formula with respect to and evaluating at , we obtain
Therefore, letting denote the bundle projection, we find that
for any elements and in where . This shows that is an isometry.
As an immediate consequence we obtain the following result.
Corollary 3.2.
The space equipped with the rightinvariant metric (\mbox{3.1}) has strictly positive constant sectional curvature equal to .
Proof.
As in finite dimensions, sectional curvature of the sphere equipped with the induced metric is constant and equal to . The computation is straightforward using for example the GaussCodazzi equations. ∎
It is worth pointing out that the bigger the volume of the manifold the bigger the radius of the sphere and therefore, by the above corollary, the smaller the curvature of the corresponding space of densities . Thus, in the case of a manifold of infinite volume one would expect the space of densities with the metric (\mbox{3.1}) to be “flat.” Observe also that rescaling the metric (\mbox{3.1}) to
changes the radius of the sphere to .
3.2. The distance and diameter of
The right invariant metric (\mbox{3.1}) induces a Riemannian distance between densities (measures) of fixed total volume on that is analogous to the Wasserstein distance (\mbox{2.14}) induced by the noninvariant metric used in the standard optimal transport. It turns out that the isometry constructed in Theorem 3.1 makes the computations of distances in with respect to (\mbox{3.1}) simpler than one would expect by comparison with the Wasserstein case.
Consider two (smooth) measures and on of the same total volume which are absolutely continuous with respect to the reference measure . Let and be the corresponding RadonNikodym derivatives of and with respect to .
Theorem 3.4.
The Riemannian distance defined by the metric (\mbox{3.1}) between measures and in the density space is
(\mbox{3.2}) 
Equivalently, if and are two diffeomorphisms mapping the volume form to and , respectively, then the distance between and is
Proof.
Let and . If and then using the explicit isometry constructed in Theorem 3.1 it is sufficient to compute the distance between the functions and considered as points on the sphere with the induced metric from . Since geodesics of this metric are the great circles on it follows that the length of the corresponding arc joining and is given by
which is precisely formula (\mbox{3.2}). ∎
We can now compute precisely the diameter of the space of densities using standard formula
Proposition 3.5.
The diameter of the space equipped with the metric (\mbox{3.1}) equals .
Proof.
First, observe that for any two densities and in we have
The arccosine of this integral is less than which yields
as an upper bound for the diameter.
In order to show that the diameter of is in fact equal to we construct a sequence of measures such that the distance between them converges to this limit. Given any large consider a disk of volume with respect to the reference measure . Let be a smooth measure whose RadonNikodym derivative is a mollification of , where is the characteristic function of the disk, see Figure 3.6. Note that the total volume of is the same as that of .
We will now estimate the distance between and . Since on the disk the function is approximately equal to , the integral appearing in the argument of arccosine in (\mbox{3.2}) can be estimated by
It follows now that as which completes the proof. ∎
Remark 3.7 (Applications to shape theory).
It is tempting to apply the distance to problems of computer vision and shape recognition.
Given a bounded domain in the plane (a 2D “shape”) one can mollify the corresponding characteristic function and associate with it (up to a choice of the mollifier) a smooth measure normalized to have total volume equal to 1. One can now use the above formula (\mbox{3.2}) to introduce a notion of “distance” between two 2D “shapes” and by integrating the product of the corresponding RadonNikodym derivatives with respect to the 2D Lebesgue measure. It is not difficult to check that when and are mutually singular and that whenever they coincide. This works also in the case when one of the measures is of deltatype and the other is very thinly distributed over a large area.
In this context it is interesting to compare the spherical metric to other rightinvariant Sobolev metrics that have been introduced in shape theory. For example, in [32] the authors proposed to study 2D “shapes” using a certain Kähler metric on the Virasoro orbits of type , see e.g., [22, 36]. This metric is particularly interesting because it is related to the unique complex structure on the Virasoro orbits. Furthermore, it has negative sectional curvature. We refer to [32] for details.
Example 3.8 (gradients on the space of densities).
The Wasserstein metric (\mbox{2.11}) induced on the space of densities was used to study certain dissipative PDE (such as the heat and porous medium equations) as gradient flow equations on , see [30, 38]. The analogous computations of the gradients on simplify due to the isometry with the round sphere discussed above.
For instance, let be a functional of the general form
where and is a smooth function on densities. Then one computes that .
Here is a quick way to see this. For a small real parameter and any meanzero function on we have
By Theorem 3.1 one can perform the calculation using the metric on the sphere and identify the variational derivative of with its gradient so that
which gives the result.
Similarly, for the functional
one obtains . Observe that in this case the associated gradient flow equation
can be interpreted as the heat equation on densities
Thus the Dirichlet functional in the rightinvariant metric on yields the same heat equation as the Boltzmann (relative) entropy functional in the Wasserstein metric. This provides yet one more relation of these two gradient approaches to the heat equation, discussed in [2].
Remark 3.9.
Alternatively, one can carry out the calculation directly in the diffeomorphism group as follows. Let be a curve in with and where and let where is the projection, see Figure 3.3. Consider a functional lifted to as an invariant functional. We seek the gradient of the functional at in the form for some function . One has
where
(\mbox{3.3}) 
After changing variables in the integrand, using (\mbox{3.1}), and projecting with the help of we obtain
which coincides with the formula obtained above directly on .
4. Probability and infinitedimensional geometric statistics
4.1. Spherical Hellinger distance
The Riemannian distance function on the space of densities introduced in Theorem 3.4 is very closely related to the Hellinger distance in probability and statistics.
Recall that given two probability measures and on that are absolutely continuous with respect to a reference probability the Hellinger distance between and is defined as
As in the case of one checks that when and are mutually singular and that when the two measures coincide. It can also be expressed by the formula
where is the socalled Bhattacharyya coefficient (affinity) used to measure the “overlap” between statistical samples, see e.g., [9] for more details.
In order to compare the Hellinger distance with the Riemannian distance defined in (\mbox{3.2}) recall that probability measures and are normalized by the condition . As before, we shall consider the square roots of the respective RadonNikodym derivatives as points on the (unit) sphere in . One can immediately verify the following two corollaries of Theorem 3.1.
Corollary 4.1.
The Hellinger distance between the normalized densities and is equal to the distance in between the points on the unit sphere .
Corollary 4.2.
The Bhattacharyya coefficient for two normalized densities and is equal to the inner product of the corresponding positive functions and in
Let denote the angle between and viewed as unit vectors in . Then we have
while