Geometry of diffeomorphism groups

# 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 volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev -metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler-Arnold 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 Fisher-Rao (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 infinite-dimensional 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, Euler-Arnold equations, optimal transport, Hellinger distance, integrable systems.

## 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 infinite-dimensional group of volume-preserving 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 Euler-Arnold or Euler-Poincaré equations.

With a few exceptions, papers on infinite-dimensional Riemannian geometry, including diffeomorphism groups, tended to focus exclusively on either strong metrics or weak metrics of -type.1 The interest in the latter has to do with the fact that such metrics often represent kinetic energies as in the case of hydrodynamics, see e.g., [4, 12, 33], or arise naturally in probability and optimal transport problems with quadratic cost functions (such as the Wasserstein or Kantorovich-Rubinstein distance); see e.g., [5, 30, 34, 38].

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 Camassa-Holm, the Hunter-Saxton or the Euler--equations.

In this paper we focus on the metrics both from a differential-geometric 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 right-invariant 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}) ⟨⟨u,v⟩⟩=14∫Mdivu⋅divvdμ.

It descends to a non-degenerate 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 infinite-dimensional 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 well-known 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 so-called Fisher-Rao (information) metric and related Chentsov-Amari -connections which have diverse applications in asymptotic statistics, information theory and quantum mechanics, see e.g., [1, 9].

We derive the Euler-Arnold equations associated to the general right-invariant metrics which include as special cases the -dimensional (inviscid) Burgers equation, the Camassa-Holm equation, as well as variants of the Euler- equation. In the particular case of the homogeneous -metric (\mbox{1.1}) the Euler-Arnold equation has the form

 (\mbox{1.2}) ρt+u⋅∇ρ+12ρ2=−∫Mρ2dμ2μ(M),

where is a time-dependent vector field on with . This equation is a natural generalization of the completely integrable one-dimensional Hunter-Saxton 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 two-dimensional Kadomtsev-Petviashvili, Ishimori, and Davey-Stewartson equations) all known integrable evolution equations are limited to one space dimension.2 We hope that the geometric properties of (\mbox{1.2}) will make it an interesting novel example in this area.

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 right-invariant (--) Sobolev metric on given at the identity by

 (\mbox{1.3}) ⟨⟨u,v⟩⟩:=a∫M⟨u,v⟩dμ+b∫M⟨δu♭,δv♭⟩dμ+c∫M⟨du♭,dv♭⟩dμ,

where are vector fields on , is the Riemannian volume form,3 is the isomorphism defined by the metric on and and are non-negative real numbers.4 We derive the Euler-Arnold equation for the metric (\mbox{1.3}) below; while a detailed study of its geometry with the attendant curvature calculations will appear in a separate publication [21].

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 Euler-Arnold 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 Fisher-Rao 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 Euler-Arnold equation and demonstrate its complete integrability, as a geodesic flow on the sphere. Since for our equation reduces to the Hunter-Saxton 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 Euler-Arnold equation for the general -- metric (\mbox{1.3}) and show that several well-known 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 right-invariant 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 Euler-Arnold equations

In this section we describe the general setup which is convenient to study geodesics on Lie groups and homogeneous spaces equipped with right-invariant metrics.

Let be a (possibly infinite-dimensional) 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 left-translation on the group while is the corresponding right-translation . 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 right-invariant (possibly weak) Riemannian metric which is determined by an inner product on the tangent space at the identity

 ⟨⟨Rη∗eu,Rη∗ev⟩⟩η=⟨⟨u,v⟩⟩,

where and . The Euler-Arnold equation on the Lie algebra for the corresponding geodesic flow has the form

with and where the bilinear operator on is defined by

Equation (\mbox{2.1}) describes the evolution in the Lie algebra of the vector obtained by right-translating 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

 dηdt=Rη∗eu,η(0)=e.
###### 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 right-invariant 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 bi-invariant with respect to the subgroup action. (If the metric in is degenerate along the subgroup then this condition reduces to the metric bi-invariance with respect to the -action, see e.g., [20]. We shall consider the general case in Section 2.4 below.) The corresponding Euler-Arnold equation is then defined similarly as long as the metric is non-degenerate 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 volume-preserving diffeomorphisms (volumorphisms) of a closed Riemannian manifold . Consider the right-invariant metric on generated by the inner product

 (\mbox{2.3}) ⟨⟨u,v⟩⟩L2=∫M⟨u,v⟩dμ.

In this case the Euler-Arnold equation (\mbox{2.1}) is the Euler equation of an ideal incompressible fluid in

 (\mbox{2.4}) ut+∇uu=−∇p,divu=0,

where is the velocity field and is the pressure function, see [3]. In the vorticity formulation the 3D Euler equation becomes

 ωt+[u,ω]=0,whereω=curlu.

Consider the right-invariant metric on given by the inner product

 ⟨⟨u,v⟩⟩H1=∫M(⟨u,v⟩+α2⟨du♭,dv♭⟩)dμ.

The corresponding Euler-Arnold equation is sometimes called the Euler– (or Lagrangian-averaged) equation and in 3D has the form

 (\mbox{2.5}) ωt+[u,ω]=0,whereω=curlu−α2curlΔu;

see e.g. [18].

Another source of examples is related to various right-invariant Sobolev metrics on the group of all circle diffeomorphisms, as well as its one-dimensional central extension, the Virasoro group. Of particular interest are those metrics whose Euler-Arnold equations turn out to be completely integrable.

On with the metric defined by the product the Euler-Arnold equation (\mbox{2.1}) becomes the (rescaled) inviscid Burgers equation

 (\mbox{2.6}) ut+3uux=0,

while the product yields the Camassa-Holm equation

 (\mbox{2.7}) ut−utxx+3uux−2uxuxx−uuxxx=0.

Similarly, the homogeneous part of the product gives rise to the Hunter-Saxton equation

 (\mbox{2.8}) utxx+2uxuxx+uuxxx=0.

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 right-invariant metric on is defined by the inner product on such functions

 ⟨⟨u,v⟩⟩˙H1=∫S1uxvxdx

and the corresponding Euler-Arnold equation is given by the Hunter-Saxton equation (\mbox{2.8}).

We also mention that if is the Virasoro group equipped with the right-invariant metric then the Euler-Arnold equation is the periodic Korteweg-de Vries equation

 ut+3uux+cuxxx=0,

which is a shallow water approximation and the classical example of an infinite-dimensional integrable system. We refer the reader to [20] for more details on these constructions.

###### Remark 2.3.

The Hunter-Saxton 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 L2-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 bundle5 with fibre and whose base is diffeomorphic to the space of normalized smooth positive densities (or, volume forms)

 Dens(M)={ν∈Ωn(M): ν>0,∫Mdν=1},

see Moser [29]. Alternatively, let denote the Radon-Nikodym derivative of with respect to the reference volume form . Then the base (as the space of constant-volume densities) can be regarded as a convex subset of the space of smooth functions on

 M={ρ∈C∞(M,R>0):∫Mρdμ=1}.

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}) Jacμ(η∘ξ)=(Jacμ(η)∘ξ)⋅Jacμ(ξ).

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}) ⟨⟨u∘η,v∘η⟩⟩L2=∫M⟨u∘η,v∘η⟩dμ=∫M⟨u,v⟩Jacμ(η−1)dμ

where and . This metric is neither left- nor right-invariant, although it becomes right-invariant when restricted to the subgroup of volumorphisms and becomes left-invariant 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 .6 Following Otto [30] one can then introduce a metric on the base for which the projection is a Riemannian submersion: vertical vectors at are those fields with , and horizontal fields are of the form for some , since the differential of the projection is where .7

The induced metric on the base is then

 (\mbox{2.11}) ⟨⟨α,β⟩⟩ρ=∫Mρ⟨∇f,∇g⟩dμ,

where and solve and with mean-zero functions and considered as elements of the tangent space at .

The geodesic equation of the metric (\mbox{2.10}) on is

 Ddtdηdt=0

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}) ∂tu+∇uu=0

and the induced geodesic equation on the quotient space reads

 (\mbox{2.13}) ∂tϕ+12|∇ϕ|2=0

where and is a smooth function.8 Note that, as with any Riemannian submersion, if the tangent vector of a geodesic is initially horizontal, then it will remain so at later times. In our situation this corresponds to the fact that vorticity is conserved so that if a solution of (\mbox{2.12}) is initially a gradient, it will always be a gradient (and implies that satisfies (\mbox{2.13})).

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}) dist2W(ν,λ)=infη∫Mdist2M(x,η(x))dμ

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 Kantorovich-Rubinstein) distance between and and is of fundamental importance in optimal transport theory.

### 2.4. Homogeneous metrics on Diff(M)/Diffμ(M)

In this section we formulate a condition under which metrics on descend to the homogeneous space of densities and describe several examples. The non-invariant metric used in optimal transport, as well as our main example, the right-invariant metric (\mbox{1.1}), both descend to the quotient. But other natural candidates, such as the right-invariant metric on or the full metric, do not.

We start with a general observation about right-invariant metrics. Let be a closed subgroup of a group .

###### Proposition 2.5.

A right-invariant metric on descends to a right-invariant metric on the homogeneous space if and only if the inner product restricted to (the orthogonal complement of ) is bi-invariant with respect to the action by the subgroup , i.e., for any and any one has

###### 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 bi-invariance 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 right-invariant, this condition reduces to one that can be checked in the tangent space at the identity.

###### Example 2.7.

The degenerate right-invariant metric (\mbox{1.1}) on descends to a non-degenerate 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 right-invariant -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 non-zero. For example, we can take to be the divergence-free part of the field and arrange for a suitable so that .

Similarly, it follows that the full metric on obtained by right-translating 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 non-invariant 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 non-invariant Sobolev metrics analogous to the non-invariant 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

 Missing or unrecognized delimiter for \right

The first term will not be affected by a volume-preserving 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 non-invariant metric (corresponding to ) is the only one descending to the homogeneous space of densities .

## 3. The ˙H1-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 right-invariant metric induced by the inner product (\mbox{1.1}), that is

 (\mbox{3.1}) ⟨⟨u∘η,v∘η⟩⟩˙H1=14∫Mdivu⋅divvdμ

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 square-integrable functions on .9 In the next section, we will show that (\mbox{3.1}) corresponds to the Bhattacharyya coefficient (also called the affinity) in probability and statistics and that it gives rise to a spherical variant of the Hellinger distance. Thus the right-invariant -metric provides good alternative notions of “distance” and “shortest path” for (smooth) probability measures on to the ones obtained from the -Wasserstein constructions used in standard optimal transport problems.

### 3.1. An infinite-dimensional sphere

We begin by constructing an isometry between the homogeneous space of densities and an open subset of the sphere of radius

 S∞r={f∈L2(M,dμ):∫Mf2dμ=r2}

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

 Φ:η↦f=√Jacμη

defines an isometry from the space of densities equipped with the -metric (\mbox{3.1}) to an open subset of the sphere of radius

 r=√μ(M)

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 orientation-preserving diffeomorphism is a strictly positive function. Next, using the change of variables formula, we find that

 ∫MΦ2(η)dμ=∫MJacμηdμ=∫Mη∗dμ=∫η(M)dμ=μ(M)

which shows that maps diffeomorphisms into . Furthermore, observe that since for any we have

 Jacμ(ξ∘η)μ=(ξ∘η)∗μ=η∗μ=Jacμ(η)μ;

it follows that is well-defined 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

 Jacη∗μ(U)=div(U∘η−1)∘ηJacμη.

Therefore, letting denote the bundle projection, we find that

 ⟨⟨(Φ∘π)∗η(U),(Φ∘π)∗η(V)⟩⟩L2 =14∫M(divu∘η)⋅(divv∘η)Jacμηdμ =14∫Mdivu⋅divvdμ=⟨⟨U,V⟩⟩˙H1,

for any elements and in where . This shows that is an isometry.

When the above arguments extend routinely to the category of Hilbert manifolds modelled on Sobolev spaces. The fact that any positive function in belongs to the image of the map follows from Moser’s lemma [29] whose generalization to the Sobolev setting can be found for example in [12]. ∎

As an immediate consequence we obtain the following result.

###### Corollary 3.2.

The space equipped with the right-invariant 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 Gauss-Codazzi 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

 b∫Mdivu⋅divvdμ

changes the radius of the sphere to .

### 3.2. The ˙H1-distance and ˙H1-diameter of Diff(M)/Diffμ(M)

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 non-invariant 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 Radon-Nikodym 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}) dist˙H1(λ,ν)=√μ(M)arccos(1μ(M)∫M√dλdμdνdμdμ).

Equivalently, if and are two diffeomorphisms mapping the volume form to and , respectively, then the -distance between and is

 dist˙H1(η,ζ)=dist˙H1(λ,ν)=√μ(M)arccos(1μ(M)∫M√Jacμη⋅Jacμζdμ).
###### 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

 rarccos(1r2∫Mfgdμ),

which is precisely formula (\mbox{3.2}). ∎

We can now compute precisely the diameter of the space of densities using standard formula

 diam˙H1Dens(M):=sup{dist˙H1(λ,ν): λ,ν∈Dens(M)}.
###### 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

 ∫M√dλdμdνdμdμ>0.

The arc-cosine of this integral is less than which yields as an upper bound for the diameter.10

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 Radon-Nikodym 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 arc-cosine in (\mbox{3.2}) can be estimated by

 ∫M√dμdμdνNdμdμ≃∫DN√f2Ndμ≃μ(M)N√N⟶0, as N→∞.

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 Radon-Nikodym 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 delta-type and the other is very thinly distributed over a large area.

In this context it is interesting to compare the spherical metric to other right-invariant 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 (˙H1-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

 H(ρ)=∫Mh(ρ)dμ

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 mean-zero function on we have

 H(ρ+ϵβ)=∫Mh(ρ+ϵβ)dμ=∫Mh(ρ)dμ+ϵ∫Mh′(ρ)βdμ+O(ϵ2).

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

 ⟨δHδρ,β⟩=ddϵH(ρ+ϵβ)∣∣ϵ=0=∫Mh′(ρ)βdμ

which gives the result.

Similarly, for the functional

 F(ρ):=12∫|∇ρ|2μ

one obtains . Observe that in this case the associated gradient flow equation

can be interpreted as the heat equation on densities

 ∂tρ=Δρ.

Thus the Dirichlet functional in the right-invariant 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}) ∂∂t∣∣t=0Jacμη(t)=(divv∘η)⋅Jacμη.

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 infinite-dimensional 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

 dist2Hel(λ,ν)=∫M(√dλdμ−√dνdμ )2dμ.

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

 distHel(λ,ν)=√2(1−BC(λ,ν))

where is the so-called 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 Radon-Nikodym 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

 BC(λ,ν)=∫M√dλdμdνdμdμ=∫Mfgdμ.

Let denote the angle between and viewed as unit vectors in . Then we have

 distHel(λ,ν)=2sin(α/2)andBC(λ,ν)=cosα,

while

 dist˙H1(λ,ν)=α=arccosBC(λ,ν).