aa r X i v : . [ m a t h . M G ] F e b ISOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES.
JAIME SANTOS-RODR´IGUEZA
BSTRACT . Let ( X, d, m ) be a metric measure space. The study of the Wasser-stein space ( P p ( X ) , W p ) associated to X has proved useful in describing severalgeometrical properties of X. In this paper we focus on the study of isometriesof P p ( X ) for p ∈ (1 , ∞ ) under the assumption that there is some characteriza-tion of optimal maps between measures, the so called Good transport behaviour GTB p . Our first result states that the set of Dirac deltas is invariant under isome-tries of the Wasserstein space. Additionally we obtain that the isometry groupsof the base Riemannian manifold M coincides with the one of the Wassersteinspace P p ( M ) under assumptions on the manifold; namely, for p = 2 that thesectional curvature is strictly positive and for general p ∈ (1 , ∞ ) that M is aCompact Rank One Symmetric Space.
1. I
NTRODUCTION AND STATEMENT OF RESULTS .The space of probability measures equipped with the Wasserstein metric reflectsseveral geometrical properties of a metric measure space ( X, d, m ) such as; com-pactness, existence of geodesics, and non-negative sectional curvature. (see forexample [1], [17]).A natural question therefore is asking whether it is possible for the Wassersteinspace to be more symmetric than the base space. Consider the following, if g : X → X is an isometry then it is easy to check that g : P p ( X ) → P p ( X ) is alsoan isometry for any p ∈ (1 , ∞ ) . Therefore M ) ⊂ Iso( P p ( X )) . So moreconcretely the question is to determine whether these two groups of isometries arethe same, in such case we will say that X is isometrically rigid.First of all, notice that for any map g : X → X, g δ x = δ g ( x ) for all x ∈ X, i.e.the pushforward of a Dirac delta is again a Dirac delta. Hence our first approachshould be to determine whether the set of Dirac deltas is invariant under isometries.Our result in this regard is: Theorem A.
Let ( X, d, m ) be a compact metric measure space with GTB p forsome p ∈ (1 , ∞ ) . Then for any isometry
Φ : P p ( X ) → P p ( X ) the set of Diracdeltas, ∆ , is invariant, i.e. Φ(∆ ) = ∆ . This result gives us as a corollary that if two compact m.m.s. ( X, d X , m ) and ( Y, d Y , n ) have isometric L p − Wasserstein spaces then X and Y must also be iso-metric (see Corollary 3.8). Date : February 18, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Wasserstein distance, isometry group.The author was supported by research grants MTM2014-57769-3-P, MTM2017-85934-C3-2-P(MINECO), ICMAT Severo Ochoa Project SEV-2015-0554 (MINECO) and a Postdoctoral Fellow-ship from the MPI Bonn.
We will need some structure on the metric measure spaces we will we workingwith, we will assume compactness, non-branching of geodesics and that the refer-ence measure m is such that the space has the so called Good transport behaviour GTB p for some p ∈ (1 , ∞ ) . Loosely speaking, this last condition requires that op-timal transports starting from absolutely continuous measures are given by a map(see Definition 2.9). This will also imply that the geodesic in P p ( X ) induced bythese transports remains inside the set of absolutely continuous measures until itreaches its endpoint. This condition was first defined in [8] by Galaz-Garc´ıa, Kell,Mondino, and Sosa. Later, it was investigated in more detail by Kell [13].The class of metric measure spaces that satisfy GTB p is quite rich. Examplesinclude Riemannian manifolds, Alexandrov spaces, non-branching M CP ( K, N ) spaces, and non-branching RCD ∗ ( K, N ) spaces.This question of determining the structure of the group of isometries of theWasserstein space is not new. It was first posed in [14] by Kloeckner in the settingof Euclidean spaces, and later for Hadamard spaces in [4] in collaboration withBertrand. In the latter isometric rigidity is proved. While in the former, more ex-otic isometries appear, being the case of the line the most interesting. (see Lemmas . , . in [14]). Some difficulties arise when working in a compact setting; mostof the machinery used previously is no longer available, for example the unique-ness of barycenters is generally no longer true in compact spaces.In order to obtain that all the isometries of the Wasserstein space come fromisometries of the space we assume an additional hypothesis. Namely we will workon Riemannian manifolds with strict positive sectional curvature. Theorem B.
Let M be a closed Riemannian manifold with strictly positive sec-tional curvature. Then it is isometrically rigid, that is, the isometry groups of M and P ( M ) coincide.It is also possible to formulate this same question for different L p − Wassersteinspaces. In the case of R and [0 , this has been done by Geh´er, Titkos and Virosztek[10]. In said paper they prove that depending on the exponent p the behaviour ofthe Wasserstein isometries can change, being the case p = 1 the most exotic. Thesame authors have also treated the cases of discrete and Hilbert spaces in [9] and[11] respectively.In this paper we can answer this question for a certain class of Riemannian man-ifolds: Compact Rank One Symmetric Spaces (CROSSes). These spaces have niceproperties (see Subsection 4.2) that give us enough information on the limitationsthat Wasserstein isometries must have. Theorem C.
Let M be a CROSS. Then for any p ∈ (1 , ∞ ) the isometry groups of M and P p ( M ) coincide.As for other works where compactness is assumed Virosztek [18] proved that inthe particular case of the sphere L − Wasserstein isometries must send Dirac deltasto Dirac deltas. Theorem A only relies on the compactness of the space as well asstructural properties of the optimal transport there (see Definition 2.9).The structure of the paper is the following: Section 2 is devoted to the presen-tation of the optimal transport problem and the existence of solutions to it. Someof the geometric properties of the Wasserstein space such as the structure of theoptimal transport when one makes certain assumptions on the reference measure m are presented as well. Section 3 contains the proof of Theorem A. And finally, in SOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES. 3
Section 4 we prove isometric rigidity in the context of positive curvature for p = 2 and in the general case p ∈ (1 , ∞ ) for CROSSes. Acknowledgements.
The author would like to express his thanks to his advisorProf. Luis Guijarro for valuable comments made during the development of thispaper as well as for his careful reading of earlier versions of this manuscript.2. P
RELIMINARIES .In this section we will review the concepts on Optimal Transport used through-out the paper as well as the notation used. Throughout the following ( M, d ) willbe a closed Riemannian manifold equipped with its usual distance. The proofs ofthe results presented in this section can be found in [1]2.1. Optimal Transport.
Let µ, ν be two probability measures supported on am.m.s. ( X, d, m ) and p ∈ (1 , ∞ ) . Kantorovich’s problem consists of minimizingthe functional:(KP) π Z d p ( x, y ) dπ ( x, y ) among all admissible measures π ∈ Adm( µ, ν ) . The set
Adm( µ, ν ) consists ofmeasures in π ∈ P ( M × M ) that have marginals µ and ν, i.e. π ( A × M ) = µ ( A ) , π ( M × B ) = ν ( B ) , ∀ A, B ∈ B ( M ) . The intuition behind admissible plans is the following: If π ∈ Adm( µ, ν ) thenfor A × B ∈ B ( M × M ) the value π ( A × B ) holds the information of how themass from A is sent to B. Observe that the functional KP is linear and that
Adm( µ, ν ) is a convex closedset (in the narrow topology). It will turn out that Kantorovich’s problem alwayshas a solution (see for example Theorem . in [1]). Measures that minimize KPwill be called optimal transports (or optimal plans). The set of optimal transportsbetween two measures µ and ν will be denoted by Opt( µ, ν ) . Given a probability measure π ∈ Adm( µ, ν ) it would be useful to determinewhether it is optimal or not. Intuitively a point ( x, y ) ∈ supp π represents themass that is sent from x to y. So if our plan is optimal then there shouldn’t way torearrange the points in supp π in a way that decreases the value of the functional.More rigorously we have: Definition 2.1.
Let p ∈ (1 , ∞ ) , we say that a set Γ ∈ X × X is p − cyclicallymonotone if for all n ∈ N and ( x , y ) , · · · , ( x n , y n ) ∈ Γ implies n X i =1 d p ( x i , y i ) ≤ n X i =1 d p ( x i , y σ ( i ) ) for all permutations σ of { , · · · , n } . It is proved in Theorem . [1] that a plan π is optimal if and only if its support supp π is p − cyclically monotone.When there exists a measurable map T : X → X such that T µ = ν and theplan ( Id, T ) µ is optimal we will say that the optimal transport is induced by amap. As a matter of fact the problem of finding a map T that is optimal is theoriginal transport problem posed by Monge (see for example Chapter of [17]). JAIME SANTOS-RODR´IGUEZ
In general it is not possible to find an optimal transport induced by such amap. Brenier (for Euclidean spaces) [7] and McCann (for Riemannian spaces)[15] showed that if one takes the starting measure µ to be absolutely continuoussuch a map exists. In the next subsection we will describe some other spaces wherethis also possible.2.2. Wasserstein space.
Let p ∈ (1 , ∞ ) , using the solutions to the Kantorovichproblem KP it is possible to define a metric on P p ( X ) . The L p − Wasserstein metric.Let µ, ν ∈ P p ( X ) then W pp ( µ, ν ) := min (cid:26) Z d ( x, y ) dπ | π ∈ Adm( µ, ν ) (cid:27) Observation 2.2.
Usually P p ( X ) denotes the space of probability measures withfinite p − moments. Since our base spaces will always be compact then it is clearevery probability measure has finite p − moments for all p. We will keep the subindexin the notation just to stress that we are considering the L p − Wasserstein metric.
Given n ∈ N we define the set of totally atomic measures: ∆ n ( X ) := (cid:26) µ ∈ P p ( X ) | µ = n X i =1 a i δ x i , x i ∈ X, n X i =1 a i = 1 , a i > (cid:27) . It is a standard result that S n ∈ N ∆ n ( X ) W p (Theorem . in [17]). If there isno confusion on which underlying space we are working with we will simplify thenotation and use instead just ∆ n . The Wasserstein space will share many geometrical properties with the basespace X the first of these is that the Wasserstein space P p ( X ) is compact if andonly if X is compact.A metric space ( X, d ) is said to be geodesic if for every x, y ∈ X there exists acurve γ : [0 , → X such that γ = x, γ = y and d ( γ s , γ t ) = | s − t | d ( γ , γ ) , s, t ∈ [0 , . The set of geodesics of X will be denoted as Geo( X ) . It will turn out that thisis sufficient for the existence of geodesics in P p ( X ) . Theorem 2.3 ( Ambrosio, Gigli [1] ). Let ( X, d ) be a geodesic space. Then theWasserstein space ( P p ( X ) , W p ) p ∈ (1 , ∞ ) is geodesic as well. Definition 2.4.
A geodesic space ( X, d ) will be said to be non-branching if themap Geo( X ) → X × Xγ ( γ , γ t ) is injective for all t ∈ (0 , . The property of being a non-branching geodesic space is inherited by the Wasser-stein space as the next result shows:
SOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES. 5
Theorem 2.5 ( Ambrosio, Gigli [1] ). Let ( X, d, m ) be a complete and separablem.m.s. Then the space ( P p ( X ) , W p ) is non-branching. Furthermore, given a ge-odesic ( µ t ) t ∈ [0 , ⊂ P p ( X ) then for every t ∈ (0 , there exists a unique optimalplan Opt( µ , µ t ) and this plan is induced by a map from µ t . Observation 2.6.
One of the immediate consequences of the previous Theorem isthat measures in ∆ n ( X ) can only be in the interior of geodesics with endpoints in ∆ ( X ) ∪ · · · ∪ ∆ n ( X ) . This will be extremely useful for determining how Wasser-stein isometries behave.
To conclude this section we will describe with more detail the assumptions wewill make on the reference measure m and the consequences they will have on thesolutions to Kantorovich’s problem. Details can be found in the paper [13] by Kelland the refrences therein. Definition 2.7.
A measure m is said to be qualitatively non-degenerate if for all R > and x ∈ X there is a function f R,x : (0 , → (0 , ∞ ) such that lim sup t → f R,x ( t ) > and for every measurable set A ⊂ B x ( R ) and all x ∈ B x ( R ) , t ∈ (0 , we have: m ( A t,x ) ≥ f R,x ( t ) m ( A ) . Where A t,x := { γ t | γ ∈ Geo( X ) , γ ∈ A, γ = x } . This kind of measures will give us some topological information on the space:
Lemma 2.8.
Let ( X, d ) a metric space and m a qualitatively non-degenerate prob-ability measure on it. Then m is doubling and X has finite Hausdorff dimension.Proof. The first affirmation is proved in Proposition . in [13], as for the second,notice that since ( X, d ) is a doubling space then its Assouad dimension (see defi-nition . in [12]) is finite. Finally this implies that the Hausdorff dimension of X must be finite. (cid:3) As for optimal transports induced by maps we will recall definition given byKell in [13]:
Definition 2.9.
A m.m.s
X, d, m is said to have good transport behaviour GTB p iffor all µ, ν ∈ P p ( X ) such that µ ≪ m then any optimal transport between µ and ν is given by a transport map. Example 2.10.
The following spaces have good transport behaviour:(1) For p = 2 essentially non-branching M CP ( K, N ) − spaces with K ∈ R , N ∈ [1 , ∞ ) . Particularly essentially non-branching CD ∗ ( K, N ) − spaces,essentially non-branching CD ( K, N ) spaces and RCD ∗ ( K, N ) − spaces.(2) p − essentially non-branching, qualitatively non-degenerate spaces (Defini-tion 2.7) for all p ∈ (1 , ∞ ) . (3) Any (locallly) doubling measure m on ( R n , d E ) or on a Riemannian man-ifold. JAIME SANTOS-RODR´IGUEZ
Definition 2.11.
Let ( X, d, m ) a m.m.s where m is qualitatively non-degenerate.We will say that it has the p − strong interpolation property (sIP p ) for some p ∈ (1 , ∞ ) if: Given any µ , µ ∈ P p ( X ) with µ ≪ m there is a unique optimaltransport, induced by a map, and the geodesic ( µ t ) t ∈ [0 , satisfies µ t ≪ m for all t ∈ [0 , . As an immediate corollary of Theorem . in [13], we have: Theorem 2.12. If ( X, d, m ) is a non-branching m.m.s. and m is qualitatively non-degenerate then GT B p and sIP p are equivalent.
3. R
ESTRICTING W ASSERSTEIN ISOMETRIES TO D IRAC DELTAS .For the remainder of this paper we will assume that our spaces satisfy the follow-ing: ( X, d, m ) is a m.m.s. such that it is compact, non-branching, m is qualitativelynon-degenerate and with GTB p for some p ∈ (1 , ∞ ) . Proposition 3.1.
Let ( X, d, m ) be a m.m.s. with GTB p and µ ∈ P p ( X ) such that µ ≪ m Then the functional: ν W pp ( µ, ν ) , is linearly strictly convex, That is, for any ν , ν ∈ P p ( X ) and t ∈ (0 , we have W pp ( µ, (1 − t ) ν + tν ) < (1 − t ) W pp ( µ, ν ) + t W pp ( µ, ν ) . Proof.
Suppose there exist η , η ∈ P p ( X ) and t ∗ ∈ (0 , such that: W pp ( µ, (1 − t ∗ ) η + t ∗ η ) = (1 − t ∗ ) W pp ( µ, η ) + t ∗ W pp ( µ, η ) . Consider now the optimal plans π ∈ Adm( µ, η ) and π ∈ Adm( µ, η ) . It isclear that (1 − t ∗ ) π + t ∗ π is an admissible plan between µ and (1 − t ∗ ) η + t ∗ η . Then: Z d p ( x, y ) d (1 − t ∗ ) π + t ∗ π = (1 − t ∗ ) Z d p ( x, y ) dπ + t ∗ Z d p ( x, y ) dπ = (1 − t ∗ ) W pp ( µ, η ) + t ∗ W pp ( µ, η )= W pp ( µ, (1 − t ∗ ) η + t ∗ η ) . And this gives us a contradiction since the plan (1 − t ∗ ) π + t ∗ π cannot be inducedby a map. (cid:3) And this lemma gives us as a Corollary:
Corollary 3.2.
For µ ≪ m , arg max( P p ( X ) ∋ ν W pp ( µ, ν )) ⊂ ∆ . Proposition 3.3.
Let Φ ∈ Iso ( P p ( X )) then the set of absolutely continuous mea-sures µ such that Φ( µ ) is also absolutely continuous is dense in P p ( X ) . Proof.
Let µ, ν ≪ m , and consider the geodesic ( µ t ) t ∈ [0 , such that µ = µ and µ = Φ − ( ν ) . Since the m.m.s. has
GTB p we have that µ t ≪ m for all t ∈ [0 , , now apply the isometry to the geodesic to obtain a new geodesic (Φ( µ t )) t ∈ [0 , , itis clear then that Φ( µ ) = ν ≪ m and so we conclude that Φ( µ t ) ≪ m for all t ∈ (0 , . Since the measures µ, ν were picked arbitrarily we obtain the thesis. (cid:3)
Remark 3.4.
Actually we have a stronger property: For any ν ∈ P p ( X ) thereexists µ ≪ m with Φ( µ ) ≪ m such that if ( µ t ) t ∈ [0 , is the unique geodesicjoining µ with ν then Φ( µ t ) ≪ m for all t ∈ [0 , . SOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES. 7
Observation 3.5.
For every x ∈ X and R > the set ∂B δ x ( R ) is compact andlinearly convex. This is just a consecuence of the fact that the optimal transportbetween a δ x and any other measure µ is given by δ x ⊗ µ. Lemma 3.6.
Let x, y ∈ X be two points such that Φ( δ x ) , Φ( δ y ) ∈ ∆ , where Φ ∈ Iso( P p ( X )) . Then there exists a geodesic γ ∈ Geo( X ) such that γ = x, γ = y and Φ( γ t ) ∈ ∆ for all t ∈ [0 , . Proof.
Let x, y ∈ X be such that Φ( δ x ) , Φ( δ y ) ∈ ∆ . Take µ ≪ m such that Φ( µ ) ≪ m which exists by Proposition 3.3. Now consider: Mid( δ x , δ y ) := e / { ( η t ) t ∈ [0 , ∈ Geo( P p ( X )) | η = δ x , η = δ y } . and notice that this set has the following properties: • Mid( δ x , δ y ) = ∂B δ x ( d ( x,y )2 ) ∩ ∂B δ y ( d ( x,y )2 ) , so by Observation 3.5 is alinearly convex and compact set. • Φ(Mid( δ x , δ y )) = Mid(Φ( δ x ) , Φ( δ y )) . Since Φ is an isometry then it is clear that: max { W pp ( µ, ν ) | ν ∈ Mid( δ x , δ y ) } = max { W pp (Φ( µ ) , ν ) | ν ∈ Mid(Φ( δ x ) , Φ( δ y )) } . And by Proposition 3.1 and the fact that both µ, Φ( µ ) ≪ m arg max (cid:0) Mid( δ x , δ y ) ∋ ν W pp ( µ, ν ) (cid:1) ⊂ ∆ arg max (cid:0) Mid(Φ( δ x ) , Φ( δ y )) ∋ ν W pp (Φ( µ ) , ν ) (cid:1) ⊂ ∆ Hence there must exists some point z ∈ X such that δ z ∈ arg max(Mid( δ x , δ y ) ∋ ν W pp ( µ, ν )) and Φ( δ z ) ∈ ∆ . (cid:3) The idea for proving Theorem A boils down to proving that there exists some set S ⊂ X with non-empty interior such that for all x ∈ S, Φ( δ x ) ∈ ∆ . Using thenObservation 2.6 gives us the result. As for how we build the set S we recall thatfrom Lemma 2.8 the Hausdorff dimension of X is finite. So then for sufficientlyenough points x , · · · , x n such that Φ( δ x i ) ∈ ∆ the geodesic convex hull of { x , · · · x n } will have non-empty interior.Given a set E ⊂ X we define the antipodal set of E as: A ( E ) := arg max( X ∋ x d ( x, E )) . Theorem A.
Let ( X, d, m ) be a compact metric measure space with GTB p forsome p ∈ (1 , ∞ ) . Then for any isometry
Φ : P p ( X ) → P p ( X ) the set of Diracdeltas, ∆ , is invariant, i.e. Φ(∆ ) = ∆ . Proof.
Take µ ≪ m such that Φ( µ ) ≪ m (such measure exists by Proposition 3.3).Then by Corollary 3.2:(1) arg max( P p ( X ) ∋ ν W pp ( µ, ν )) ⊂ ∆ (2) arg max( P p ( X ) ∋ ν W pp (Φ( µ ) , ν )) ⊂ ∆ Since W pp ( µ, ν ) = W pp (Φ( µ ) , Φ( ν )) the isometry Φ sends the set 1 to the set 2.That is, there exists some x ∈ X such that Φ( δ x ) ∈ ∆ . Suppose now that we have found x , · · · , x n ∈ X such that Φ( δ x i ) ∈ ∆ . ByLemma 3.6 the geodesic convex hull S ( x , · · · , x n ) ⊂ X consists of points withthe property that for all y ∈ S ( x , · · · , x n ) , Φ( δ y ) ∈ ∆ . JAIME SANTOS-RODR´IGUEZ
Now consider the totally atomic measure P ni =1 1 n δ x i , by the density stated inProposition 3.3 we can find some measure µ n +1 such that: • µ n +1 , Φ( µ n +1 ) ≪ m . • supp µ n +1 ⊂ ⊔ ni =1 B x i ( ǫ i ) , for some ǫ i > , i ∈ { , · · · , n } . • There exists some y ∈ A ( S ( x , · · · , x n )) such that W pp ( µ n +1 , δ y ) ≥ W pp ( µ n +1 , δ z ) for all z ∈ S ( x , · · · , x n ) . So we just look at the arguments of the maxima for the linearly strictly con-vex functionals W pp ( µ n +1 , · ) , W pp (Φ( µ n +1 ) , · ) and obtain a point x n +1 ∈ X − S ( x , · · · , x n ) such that Φ( x n +1 ) ∈ ∆ . Since by Lemma 2.8 the Hausdorff dimension of X is finite we have that theremust exist m ∈ N such that S ( x , · · · , x m ) has nonempty interior.So for points z ∈ X and x in the interior of S ( x , · · · , x m ) a geodesic γ with endpoints y and x must satisfy: There exists t ∈ (0 , such that γ t ∈ S ( x , · · · , x m ) . So by applying Observation 2.6 then Φ( δ z ) ∈ ∆ . (cid:3) And with this we also have as a Corollary:
Corollary 3.8.
Let ( X, d X , m ) , ( Y, d Y , n ) be two compact m.m.s such that theyhave GTB p for some p ∈ (1 , ∞ ) . Suppose there exists some isometry
Φ : P p ( X ) → P p ( Y ) . Then X and Y are isometric.Proof. Let
Φ : P p ( X ) → P p ( Y ) be the isometry. Then from Theorem A we havethat for all x ∈ M, Φ( δ x ) ∈ ∆ ( Y ) . So we define F : X → Y as the function suchthat F ( δ x ) := Φ( δ x ) . It is clear that this defines an isometry of the base metricspaces. (cid:3)
4. I
SOMETRIC RIGIDITY .This last section deals with answering affirmatively the question of isometricrigidity, at least in some cases.4.1.
Positively curved manifolds and p = 2 . In this subsection we will provethat if we additionally ask that the manifold has positive sectional curvature thenit is possible to obtain isometric rigidity. We will restrict only to the case where p = 2 , the reason being that the flatness condition (see Definition 4.2) is of no useto us for general p. In the next subsection however, we will prove isometric rigidityin the case that we have more structure on the manifold.
Theorem B.
Let M be a closed Riemannian manifold of strictly positive sectionalcurvature. Then, the Wasserstein space ( P ( M ) , W ) is isometrically rigid, i.e. Iso( M ) = Iso P ( M ) . Given an isometry
Φ : P ( M ) → P ( M ) we have obtained from Theorem Athat it restricts to an isometry of the Riemannian manifold. Therefore from hereon out we will only consider isometries Φ of P ( M ) such that Φ( δ x ) = δ x for all x ∈ M. Our aim will be to prove that for all n ∈ N , µ ∈ ∆ n Φ( µ ) = µ. Since the unionof these sets over all n is dense then we will be done.Consider γ : [0 , → M a geodesic. Then γ : ( P ([0 , , W ) → ( P ( M ) , W ) is an isometric embedding of ( P ([0 , , W ) into ( P ( M ) , W ) . We will deontethe set of probability measures supported in the geodesic γ as P ( γ ) . SOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES. 9
Our first step will be to prove that P ( γ ) is not only invariant under isometries Φ that fix Dirac deltas but that Φ( µ ) = µ for all µ ∈ P ( γ ) . In general, regardless on any curvature assumption on the manifold M the onlypossible totally geodesic embedded submanifolds that one can expect to find areprecisely the minimizing geodesics. These geodesics are actually flat spaces in thesense that their curvature is identically 0. The next definition gives an alternativeformulation of a metric space being flat just in terms of the distance. Definition 4.2.
Let ( X, d ) be a geodesic space, we will say that it is flat if givenany three points x, y, z ∈ X and every γ : [0 , → X geodesic such that γ = y, γ = z we have: d ( x, γ t ) = (1 − t ) d ( x, γ ) + td ( x, γ ) − (1 − t ) td ( y, z ) , ∀ t ∈ [0 , . Examples of flat spaces include Hilbert spaces, closed intervals [ a, b ] equippedwith an interior product, and Wasserstein spaces ( P ([ a, b ]) , W ) . One importantobservation to make though is that even if the base space X is flat then P ( X ) ingeneral is only non-negatively curved. (see Example . . in [1]). Observation 4.3.
Before moving on with the next results we will make some ob-servations regarding the structure of the Wasserstein space of a closed interval say, [0 , equipped with its usual Euclidean metric. In [14] it is noted (Proposition . ) that the set ∆ ( R ) plays a special role as the convex hull of it is dense. Thisis used in order to define the exotic isometries (see Lemma . in [14] ). It is clearthat the convex hull of ∆ ([0 , will also be dense in P ([0 , . We will use this as well in the proofs of the next Propositions. The next resultappears in Section . of [10], we include a proof since our arguments are different. Proposition 4.4.
Let ([0 , , d E ) be the interval equipped with its usual Euclideanmetric. Then the Wasserstein space ( P ([0 , , W ) is isometrically rigid.Proof. First, let us notice the following: For the functional µ W ( δ / , µ ) , µ ∈ P ([0 , (cid:0) µ W ( δ / , µ (cid:1) = { (1 − λ ) δ + λδ | λ ∈ [0 , } . We can mimic the argument done in Theorem A to obtain that all Wassersteinisometries send Dirac deltas to Dirac deltas. Hence it suffices to look at isometries
Φ : P ([0 , → P ([0 , such that Φ | δ ≡ id. It is immediate that
Φ ( { (1 − λ ) δ + λδ | λ ∈ [0 , } ) = { (1 − λ ) δ + λδ | λ ∈ [0 , } since Φ( δ / ) = δ / . Now, for (1 − λ ) δ + λδ suppose Φ((1 − λ ) δ + λδ ) = (1 − t ) δ + tδ forsome t ∈ (0 , . Then t = W ((1 − t ) δ + tδ , δ ) = W ((1 − λ ) δ + λδ , δ ) = λ. So t = λ. That is Φ restricted to { (1 − λ ) δ + λδ | λ ∈ [0 , } is the identity.Consider now (1 − λ ) δ a + λδ b , WLOG a < b, then it is an interior point of theunique geodesic joining δ a/ (1+ a − b ) with (1 − λ ) δ + λδ . Since Φ fixes both δ a/ (1+ a − b ) and (1 − λ ) δ + λδ it must fix the whole geodesicincluding (1 − λ ) δ a + λδ b (See Observation 2.6), hence Φ fixes ∆ ([0 , . UsingObservation 4.3 we conclude then that Φ ≡ id. (cid:3) Proposition 4.5.
Let x ∈ M and consider γ a geodesic starting at x and such thatit cannot be extended past γ while remaining minimizing. Then for any isometry Φ such that it fixes all Dirac deltas we have that Φ( P ( γ )) = P ( γ ) . Proof.
Let
Φ : P ( M ) → P ( M ) be an isometry such that Φ | ∆ ≡ id. In or-der to get the result, from the Observation 4.3 it will be sufficient to prove that
Φ(∆ ( γ )) = ∆ ( γ ) . First, we will prove that the set { (1 − λ ) δ γ + λδ γ | λ ∈ [0 , } is invariant.Consider two points in γ ([0 , and a geodesic σ : [0 , → M joining them.Notice that σ is completely contained in γ ([0 , . Let µ = (1 − λ ) δ γ + λδ γ , since P ( γ ) is a flat space we have that W ( µ, δ σ t ) = (1 − t ) W ( µ, δ σ ) + t W ( µ, δ σ ) − (1 − t ) t W ( δ σ , δ σ ) , for all t ∈ [0 , . And since Φ fixes every Dirac delta we have an analogous equationfor Φ( µ ) , which can be rewritten in the following way since the product measure Φ( µ ) ⊗ δ σ t is optimal for all t ∈ [0 , . Z d ( y, σ t ) − (1 − t ) d ( y, σ ) − td ( y, σ ) + t (1 − t ) d ( σ , σ ) d Φ( µ )( y ) = 0 . As the manifold is of positive sectional curvature then the integrand must the Φ( µ ) − a.e. identically 0. That is, d ( y, σ t ) − (1 − t ) d ( y, σ ) − td ( y, σ ) + t (1 − t ) d ( σ , σ )) = 0 , for all t ∈ [0 , and Φ( µ ) -a.e. The positive curvature then forces the support of Φ( µ ) to be in γ [0 , , otherwise we would have Euclidean traingles embedded in M. Therefore we obtain the thesis. (cid:3)
Clearly given a geodesic γ ∈ Geo( M ) we have that P ( γ ) and P ([0 , d ( γ , γ )]) are isometric so combining Propositions 4.4 and 4.5 we obtain the following corol-lary: Corollary 4.6.
For any geodesic γ : [0 , → M and Φ : P ( M ) → P ( M ) anisometry that fixes all Dirac deltas. Φ restricted to ( P ( γ ) , W ) is the identity map. And noting that given any µ ∈ ∆ ( M ) its atoms are contained in some geodesicwe obtain: Corollary 4.7.
Let
Φ : P ( M ) → P ( M ) be an isometry such that it fixes Diracdeltas. Then Φ fixes ∆ ( M ) as well. Now we will like to see what happens when we look at measures not neces-sarily supported on a geodesic. Given a measure µ ∈ P ( M ) and a geodesic γ ∈ Geo( M ) we define proj γ ( µ ) the projection of µ onto γ as:(3) proj γ ( µ ) = arg min (cid:0) ν ∈ P ( γ ) W ( µ, ν ) (cid:1) . Note that in general µ proj γ ( µ ) is not a function since the set on right handside of 3 may contain more than one element. For example consider in the sphere µ = δ N the north pole and γ a geodesic in the equator. It is clear that every measurein P ( γ ) is equidistant to ∆ N . Nevertheless it will be very useful to us to work with projections onto geodesics.It is easy to convince oneself that if µ is a totally atomic measure in say, ∆ n ( M ) the projection onto any geodesic contains at least one totally atomic measure. SOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES. 11
Proposition 4.8.
Let µ ∈ ∆ n ( M ) , then for every geodesic γ then proj γ ( µ ) ∩ (∆ ( M ) ∪ · · · ∆ n ( M )) = ∅ . Proof.
Let µ = P ni =1 λ i δ x i and γ ∈ Geo( M ) . Take now a measure ν ∈ P ( γ ) . Take π ∈ Opt( µ, ν ) , so then: W ( µ, ν ) = Z M × M d ( x, y ) dπ ( x, y )= Z { x }× M d ( x, y ) dπ ( x, y ) + · · · + Z { x n }× M d ( x, y ) dπ ( x, y ) ≥ λ d ( x , y ) + · · · + λ n d ( x n , y n ) . Where the points y i are such that: y i ∈ arg min (cid:0) ∆ ( γ ) ∋ δ y d ( x i , y ) (cid:1) , ∀ ≤ i ≤ n. Hence we conclude that n X i =1 λ i δ y i ∈ proj γ ( µ ) . (cid:3) So, let us describe our plan for proving the isometric rigidity. First we willprove that for each n ∈ N the set ∆ n ( M ) is invariant. Then we will prove thattotally atomic measures supported on a sufficently small ball B are fixed. A densityargument will yield that any measure whose support is contained in B is also fixed.Finally we will use a non-branching argument to conclude.We divide each of these steps into several Lemmas to make the argument asclear as posible. Lemma 4.9.
Let
Φ : P ( M ) → P ( M ) be an isometry that fixes all Dirac deltas,consider γ ∈ Geo( M ) and P ( γ ) the set of measures supported at γ. For ǫ ≪ consider: Tub ǫ ( P ( γ )) := [ µ ∈ P ( γ ) B µ ( ǫ ) a tubular neighbourhood around P ( γ ) . Then for every n ∈ N , and µ ∈ ∆ n ( M ) such that supp( µ ) ⊂ Tub ǫ ( P ( γ )) we have that Φ( µ ) ∈ ∆ n ( M ) . Proof.
We may assume that all the measures considered are such that they givezero mass to γ [0 , . The proof will be done by induction over n ∈ N . For n = 1 the result is clear as it follows from Theorem A. Let µ ∈ ∆ n +1 ( M ) be such that supp( µ ) ⊂ Tub ǫ ( P ( γ )) . Then we may write µ = P n +1 i =1 λ i δ x i , where all atomsare different and for all i, λ i = 0 . Now take ν ∈ proj γ ( µ ) ∩ ∆ n +1 ( γ ) such that ν = P n +1 i =1 λ i δ y i . Denote by r i = d ( x i , y i ) . Notice that since ǫ is sufficiently small there exists only one geodesicbetween x i and y i , and that such geodesic may be extended up to another point z i such that d ( x i , z i ) = 2 r i . Note that this makes ν the midpoint between µ and the totally atomic measure P n +1 i =1 λ i δ z i . And so we have that since Φ( ν ) = ν using Theorem 2.5 and theinduction hypothesis this forces Φ( µ ) ∈ ∆ n +1 ( M ) . (cid:3) Lemma 4.10.
Let
Φ : P ( M ) → P ( M ) be an isometry that fixes all Dirac deltas,then for every n ∈ N Φ(∆ n ( M )) = ∆ n ( M ) . Proof.
Let µ ∈ ∆ n ( M ) , and consider a geodesic γ ∈ Geo( M ) such that for some ν ∈ proj γ ( µ ) ∩ ∆ n ( γ ) . Furthermore assume that the transport between these twomeasures is given by a map.Now if we consider the Wasserstein geodesic ( η t ) t ∈ [0 , ⊂ ∆ n ( M ) between Φ( µ ) and ν = Φ( ν ) there exists some t ∈ (0 , such that supp( η t ) ⊂ Tub ǫ ( P ( γ )) for some sufficiently small < ǫ. From the previous Lemma 4.9 we obtain that η t ∈ ∆ n ( M ) , and from Theorem 2.5 we have that Φ( µ ) ∈ ∆ n ( γ ) . (cid:3) Remark 4.11.
Notice that in the proofs of the previous Lemmas 4.9, 4.10 thetransports considered were actually given by a map. Hence the weights given toeach of the atoms are also preserved.
Lemma 4.12.
Let
Φ : P ( M ) → P ( M ) be an isometry that fixes all Dirac deltas,and n ∈ N then for every µ = P ni =1 λ i δ x i ∈ ∆ n ( M ) such that d ( x i , x j ) < ǫ forall i = j and ǫ sufficently small we have that Φ( µ ) = µ. Proof.
Given a geodesic γ ∈ Geo( M ) it is clear that proj γ ( µ ) consists only oftotally atomic measures. It also easy to check that: proj γ ( µ ) = Φ(proj γ ( µ )) = proj γ (Φ( µ )) . Furthermore if we additionally assume that γ t = x for some t ∈ [0 , we obtainthat for all ν ∈ proj γ ( µ ) , ν ( { x } ) ≥ λ . From Lemma 4.10 we know that Φ( µ ) is also totally atomic, actually with thesame number of atoms as µ. Suppose that there exists some r > such that d ( x , y ) > r for all y atom of Φ( µ ) . This implies however, that there exists somegeodesic σ ∈ Geo( M ) such that σ = x and that for some ˜ ν ∈ proj σ (Φ( µ ))˜ ν ( { x } ) = 0 . This gives us a contradiction. Hence x is one of the atoms of Φ( µ ) . We repeat this argument for every x i and conclude that µ and Φ( µ ) must havethe same atoms. From the Remark 4.11 we conclude then that µ = Φ( µ ) . (cid:3) Lemma 4.13. M is isometrically rigid.Proof. Take a fixed point w ∈ M and some ǫ ≪ . from the Lemma 4.12 we havethat for all µ ∈ P ( B w ( ǫ )) , Φ( µ ) = µ. Consider then µ P ( B w ( ǫ )) absolutely continuous and ν ∈ P ( M ) some arbi-trary measure. Additionally assume that supp( µ ) ⊂ B w ( ǫ/ . Let ( η t ) t ∈ [0 , bethe unique Wasserstein geodesic such that η = µ and η = ν. Hence it followsthat (Φ η t ) t ∈ [0 , is also a Wasserstein geodesic but now with endpoints µ and Φ( ν ) . Then there exists some t ∈ (0 , such that supp Φ( η t ) ⊂ B w ( ǫ ) , hence Φ( η t ) = η t . Since the Wasserstein space P ( M ) is non-branching it followsthen that Φ( ν ) = ν. Therefore M is isometrically rigid. (cid:3) Rigidity on CROSSes.
In this last subsection we will restrict ourselves to theclass of Compact Riemannian Symmetric Spaces (CROSSes). Several properties ofthese spaces are discussed in Chapter of [6].They have been completely classifiedand are: Euclidean spheres, projective spaces (with field either real, complex orquaternionic numbers), and the Cayley plane.In the next Lemma we summarize the properties that will use: Lemma 4.14.
Let M be a CROSS. Then: • For any point x ∈ M the isotropy group at x acts transitively on any sphere ∂B x ( R ) . SOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES. 13 • For any x ∈ M the cut locus, Cut( x ) , is either a point or a totally geodesicembedded CROSS of codimension . An immediate but important consecuence of this Lemma is that for any point x the distance to Cut( x ) is constant and equal to the diameter of M. Using Theorem A we will now restrict to isometries that fix every Dirac delta.But before continuing let us describe the motivation of our strategy to prove The-orem C. In [14] Kloeckner proved (Proposition . ) that for n ≥ the geodesicconvex hull of ∆ ( R ) is dense in P ( R ) . Thefore it is enough to describe the be-haviour of the isometries on ∆ ( R ) . Now, in our setting, fix some point x ∈ M and consider the Lipschitz function: d ( x, · ) : M → [0 , Diam( M )] . The preimage of an element in ∆ n ([0 , Diam( M )]) consists of measures whosesupports are contained on spheres ∂B x ( r i ) 1 ≤ i ≤ n ; for example, measuresin ∆ n ( M ) are included here. Therefore a naive (but ultimately useful) approachwould be to look at the behaviour of Wasserstein isometries at these measures.More precisely we have: Proposition 4.15.
Let M be a CROSS, Φ ∈ Iso( P p ( X )) , p ∈ (1 , ∞ ) and assumethat for all x ∈ M every probability measure supported on Cut( x ) is fixed by Φ . Then, for µ ∈ P p ( M ) such that: supp( µ ) ⊂ n [ i =1 ∂B x ( r i ) , ≤ r i ≤ d ( x, Cut( x )) we have that supp(Φ( µ )) ⊂ n [ i =1 ∂B x ( r i ) , ≤ r i ≤ d ( x, Cut( x )) . Moreover, the weights are preserved, i.e. µ ( ∂B x ( r i )) = Φ( µ )( ∂B x ( r i )) for all i ∈ { , · · · , n } . Before proving this Proposition we will need a couple simple observations andan auxiliary lemma:
Observation 4.16.
The following are simple properties of geodesics in the Wasser-stein space P p ( M ) . • If µ , µ ∈ P p ( M ) are fixed by Φ ∈ Iso( P p ( M )) then the set of geodesicswith endpoints µ and µ is invariant under Φ . • If µ = δ x , and µ such that supp µ ⊂ Cut( x ) . Then for any ( η ) t ∈ [0 , ∈ Geo( P p ( X )) joining them we have that supp η t ⊂ ∂B x ( td ( x, Cut( x ))) for all t ∈ [0 , . • If µ = δ x and ν is supported on some ∂B x ( td ( x, Cut( x ))) , < t < then there exists a measure µ supported on Cut( x ) such that ν is in theinterior of some geodesic joining µ and µ . This is clear, just notice thatevery point in the support of ν is of the form γ t for some geodesic startingat x. So just extending these geodesics yields the measure µ . Lemma 4.17.
Let M be a CROSS, Φ ∈ Iso( P p ( M )) , p ∈ (1 , ∞ ) and assume thatfor all x ∈ M every probability measure supported on Cut( x ) is fixed by Φ . Then,for µ ∈ P p ( M ) such that: supp( µ ) ⊂ n [ i =1 ∂B x ( r i ) , ≤ r i ≤ d ( x, Cut( x )) . Then • For n = 2 there exist ν supported on { x } ∪ Cut( x ) and ν supported onsome ∂B x ( r ) such that the geodesic between them passes through µ. • If n ≥ there exist ν , ν ∈ P p ( X ) supported on n − spheres ∂B x ( r i ) and such that the geodesic between them passes through µ. Proof.
It will be sufficient to consider totally atomic measures µ such that theysatisfy the following: If y ∈ supp µ and γ ∈ Geo( X ) is such that γ = x and γ t = y for some t ∈ [0 , then γ [0 , ∩ supp µ = { y } . Let D denote the diameterof M. Case ( n = 2 ). Consider µ = (1 − λ ) µ + λµ where supp µ i ⊂ ∂B x ( r i ) and r < r < D. As µ is supported on a single sphere then from Observation 4.16we can find some measure ˜ µ, with supp ˜ µ ⊂ Cut( x ) such that µ is in the interiorof some geodesic joining δ x with ˜ µ. We will take then ν = (1 − λ ) δ x + λ ˜ µ. As for ν we do the following: Every point in supp µ i is of the form γ t i , where t i = r i /D,γ ∈ Geo( M ) starting at x and i = { , } . So we can just send the mass of thesepoints to γ t / (1+ t − t ) . Therefore the measure ν will be a totally atomic measure whose atoms are ofthe form γ t / (1+ t − t ) with mass either µ ( γ t ) > or µ ( γ t ) > . It is easy tosee that the geodesic joining ν with ν passes through µ. Case ( n ≥ ). Let now µ = a µ + · · · + a n µ n , supp µ i ⊂ ∂B x ( r i ) , a i > , and < r < · · · < r n < D. We will do a similar construction as in the previous case.Take ν = a µ + a η + a µ + · · · + a n µ n ,ν = a µ + a ˜ η + a µ + · · · + a n µ n . Where supp η ⊂ ∂B x ( r ) , supp ˜ η ⊂ ∂B x ( r ) . These measures η, ˜ η are obtainedby sending the mass of µ along the appropiate geodesics to ∂B x ( r ) and ∂B x ( r ) respectively. It is clear that µ is in the interior of the geodesic between ν and ν (cid:3) With this previous result we can now prove the Proposition.
Proof of Proposition 4.15.
We will proceed by induction on the dimension of M. Take x ∈ M, since Cut( x ) is either a CROSS or a point then we can assume thatthe measures supported there are fixed by Φ . Denote by D the diameter of M. Take µ ∈ P p ( X ) and let n bethe number of spheres ∂B x ( r i ) on which thesupport of µ is contained. The case n = 1 follows clearly from Observation 4.16. Case ( n = 2 ). Consider a measure of the form (1 − λ ) δ x + λµ where µ is sup-ported on Cut( x ) and absolutely continuous with respect to the volume measureof Cut( x ) . SOMETRIC RIGIDITY OF COMPACT WASSERSTEIN SPACES. 15
Observe that the geodesic from µ to (1 − λ ) δ x + λµ cannot be extended furthersince the length of the geodesics involved in the optimal transport is already max-imal. This implies that there must exists some set Γ ⊂ supp π, π ∈ Opt ( µ, (1 − λ ) δ x + λµ ) such that π (Γ) > and for all ( y, z ) ∈ Γ , d ( y, z ) = D. Also, fromthe p − cyclical monotone condition (see Definition 2.1) for all ( y , z ) , ( y , z ) d ( y , z ) = d ( y , z ) = D. Now, since µ ( e Γ) > and µ is absolutely continuous with respect to the vol-ume measure on Cut( x ) it follows that e Γ = { x } . So we deduce that
Φ((1 − λ ) δ x + λµ ) = (1 − α ) δ x + αν, α ∈ (0 , and ν ∈ P p ( M ) . Consider now η ∈ Mid( δ x , µ ) , so W pp (1 − α ) δ x + αν, η ) ≤ (1 − α ) W pp ( δ x , η ) + α W pp ( ν, η )= (1 − α )( D p + α W pp ( ν, η ) . If there exists a set ¯Γ ⊂ ¯ π, π ∈ Opt( η, ν ) such that ¯ π (¯Γ) > and for all ( y, z ) ∈ ¯Γ d ( y, z ) < ( D/ p then W pp ( ν, η ) < ( D/ p . But this contradicts the fact that W pp ((1 − λ ) δ x + λµ, Φ − ( η )) = ( D/ p . As before, the p − cyclical monotonicity of the support of ¯ π guarantees that forevery z ∈ supp ν d ( x, z ) = D, i.e. supp ν ⊂ Cut( x ) . Finally, W pp ((1 − λ ) δ x + λµ, µ ) = (1 − λ ) D p , W pp ((1 − λ ) δ x + λµ, δ x ) = λD p forces α = λ and ν = µ. The case proved just now is sufficient. First, from the density of the absolutelycontinuous measures supported on
Cut( x ) we can extend it for all µ ∈ P p ( M ) , with supp µ ⊂ Cut( x ) . Next, given a measure ν supported on ∂B x ( r ) ∪ ∂B x ( r ) , < r < r < D there exists by Lemma 4.17 measures µ , µ supported on { x } ∪ Cut( x ) , ∂B x ( r / (1 + r − r )) respectively such that ν is in the interiorof the geodesic joining µ with µ . As previously proved both Φ( µ ) , Φ( µ ) aresupported on the same spheres and this forces Φ( ν ) to do so as well. Case ( n ≥ ). By an induction argument on n we obtain the thesis. Lemma4.17 tell us that measures µ supported on n spheres lie in the interior of a geodesicjoining two measures supported on n − spheres. So by the induction hypothesisthe endpoints when we apply the isometry Φ are supported on the same spheres.Hence this also happens to the support of Φ( µ ) . (cid:3) And so finally we prove the main Theorem of this subsection.
Theorem C.
Let M be a CROSS. Then for any p ∈ (1 , ∞ ) the isometry groups of M and P p ( M ) coincide.Proof. Let us do induction on the dimension of the space M. Take µ ∈ ∆ n , suchthat µ = P ni =1 a i δ x i where a i > , P ni =1 a i = 1 . Fix x and notice that Cut( x ) is either a point or a totally geodesic embedded CROSS of dimension one lessthan the dimension of X. By the induction hypothesis we have that any probabilitymeasure supported on
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