Isometrically Flexible Wasserstein Spaces

• Isometrically flexible Wasserstein spaces are metric measure spaces exhibiting additional isometries—beyond push-forwards of base space isometries—that lead to exotic mass-splitting and internal rotations.
• Hilbertian splitting in Euclidean and Hilbert spaces for p=2 enables novel actions like rotations about the barycenter and fiberwise transformations, highlighting a departure from traditional rigidity.
• The sensitivity of Wasserstein space isometries to metric perturbations shows that minor augmentations can restore rigidity, underscoring a delicate balance in the geometry of optimal transport.

Isometrically flexible Wasserstein spaces are metric measure spaces whose $p$-Wasserstein isometry group exceeds that induced by the isometries of the underlying metric space. This phenomenon is exceptional in the landscape of optimal transportation, in which rigidity—meaning all isometries come from push-forwards by base isometries—is generic. Flexibility instead marks the presence of exotic, typically measure-shape-altering or mass-splitting isometries, or higher-dimensional internal symmetries, in the Wasserstein space geometry. The characterization, construction, and implications of such flexibility are central in understanding the geometric and group-theoretic structure of Wasserstein spaces.

1. Definitions and Classification of Rigidity vs Flexibility

Let $(X, d)$ be a Polish metric space and fix $p > 0$. The $p$-Wasserstein space is $(\mathcal{P}_p(X), W_p)$, where $W_p$ is the $p$-Wasserstein distance defined by the usual infimal transport cost over $\Pi(\mu, \nu)$, the space of couplings between $\mu$ and $\nu$.

Isometrically rigid: Every isometry $\Phi \colon \mathcal{P}_p(X) \to \mathcal{P}_p(X)$ arises as push-forward by some isometry $\varphi \in \operatorname{Isom}(X)$, so $\Phi = \varphi_{\#}$.

Isometrically flexible: There exist isometries of $\mathcal{P}_p(X)$ that are not push-forwards by isometries of $X$, i.e., $$\operatorname{Isom}(\mathcal{P}_p(X)) \supsetneq \{\varphi_{\#} : \varphi\in\operatorname{Isom}(X)\}$$.

The phenomenon of flexibility typically manifests in Euclidean spaces with $p=2$ or on select metric spaces with adjusted metrics/constructions, and may take the form of "internal" rotations, involutive flips, or mass-splitting isometries (Gehér et al., 2021, Balogh et al., 13 Feb 2025).

2. Origins and Prototypical Constructions of Flexible Wasserstein Structures

2.1. Hilbertian Splitting and Internal Symmetries

The seminal case of flexibility arises for $p=2$ over Euclidean or Hilbert spaces. For $X = \mathbb{R}^n$ and $p=2$ (classically, $W_2(\mathbb{R}^n)$), the isometry group strictly contains the push-forwards by Euclidean isometries. An explicit example is the action of orthogonal transformations about the barycenter ("rotate-around-centre-of-mass"):

$$\Phi_R \colon \mathcal{P}_2(\mathbb{R}^n) \to \mathcal{P}_2(\mathbb{R}^n), \qquad \Phi_R(\mu) = \left(x \mapsto R(x - m(\mu)) + m(\mu)\right)_\# \mu$$

where $m(\mu)$ is the barycenter and $R \in O(n)$.

Generalizations to $H \times Y$, with $H$ a Hilbert space and $Y$ any proper metric space, yield further exotic isometries, such as "fiberwise" rotations (Che et al., 2024). These actions cannot be realized by any isometry of the base space and demonstrate deep Hilbertian splitting at the measure level.

2.2. The Mass-Splitting/Flip Phenomenon on $[0,1]$

For $p=1$ over $[0,1]$, the flip map $j$ is an isometry given by

$F_{j(\mu)} = F_\mu^{-1}$

where $F_\mu$ is the CDF of $\mu$. The map $j$ does not preserve the class of Dirac measures: $j(\delta_t) = t\delta_0 + (1-t)\delta_1$ is a genuine example of a mass-splitting isometry (Gehér et al., 2020, Gehér et al., 2021). The resulting isometry group for $W_1([0,1])$ is the Klein four group, generated by $j$ and the reflection.

More generally, the construction of product metric spaces $Y = [0,1] \times X$ with $d_Y((t,x),(t',x')) = (|t-t'| + d_X(x,x')^q)^{1/q}$ for $q\geq 1$ yields Wasserstein spaces $W_p(Y)$ with mass-splitting isometries supported by extending the flip map fiberwise (Balogh et al., 13 Feb 2025).

2.3. Product and Discrete Spaces

Even $\mathcal{W}_p(\mathcal{X})$ for $X$ equipped with the discrete metric shows remarkable flexibility at the non-surjective, embedding level; isometric embeddings can split atoms arbitrarily among disjoint "clouds" associated to the base points (Gehér et al., 2018). However, only surjective isometries retain rigidity.

3. Geometric and Algebraic Characterizations

Isometric flexibility in Wasserstein spaces is tightly linked to geometric features of the base space and the convexity or concavity properties of the transport cost function.

• For $p<1$, strict concavity forces any isometry to fix Dirac masses, essentially precluding flexibility.
• For $p=1$, flexibility occurs in certain ultrametric or degenerate interval spaces but not in settings with strict triangle inequalities.
• For $1<p<\infty$, the non-quadratic cost typically enforces rigidity except in Hilbertian settings with $p=2$.

In the infinite-dimensional Hilbert case, the classification is exhaustive: only for $p=2$ is there extra flexibility, whereas $p\neq 2$ admits only push-forward isometries (Gehér et al., 2021).

A pivotal criterion is Hilbertian splitting: if $X = H \times Y$ with $H$ Hilbert, $W_2(X)$ splits accordingly, facilitating internal exotic actions and flexibility (Che et al., 2024).

Base spaces with negative curvature (Hadamard spaces $X$ with CAT($\kappa<0$) and visibility) confer strong rigidity on $W_2(X)$. Any isometry of $W_2(X)$ must arise from an isometry of $X$ (Bertrand et al., 2014, Bertrand et al., 2010). Conversely, Euclidean bases with $p=2$ are flexible.

Table: Rigidity vs. Flexibility by Base and Exponent

Space/Exponent	$p = 1$	$1 < p < 2$	$p=2$
Hilbert splitting $H \times Y$	?	?	Flexible
Ray $[0, \infty)$	Rigid	Rigid	Rigid
Spherical suspensions ($\dim<\pi/2$)	?	Rigid	Rigid
$[0,1]$ (interval)	Flexible	Rigid	Rigid
$\mathbb{R}$	Rigid	Rigid	Flexible

(Che et al., 2024, Gehér et al., 2021, Gehér et al., 2020)

4. Destruction and Restoration of Flexibility

A characteristic of isometric flexibility in Wasserstein spaces is its instability under mild perturbations:

• Rigidity by augmentation: Adjoining a single "far" point $q$ to $[0,1]$ (or to $\mathbb{R}^2$) at large distance $D$ immediately restores rigidity to $W_1([0,1]\cup\{q\})$ ($D>1$), or $W_2(\mathbb{R}^2 \cup \{q\})$—trivial isometries only survive. Each isometry must preserve the slice structure induced by the measure of $\{q\}$, precluding mass-splitting/flipping (Balogh et al., 26 Jan 2026).
• Metric and exponent sensitivity: Adjusting the product metric on $Y=[0,1]\times X$ from $\ell^1$ to an $\ell^\alpha$ with $\alpha<1$ can induce rigidity (Balogh et al., 13 Feb 2025).

This instability marks flexibility as a phenomenon of finely tuned geometric or analytic situations; generic perturbations of the base or cost function kill it.

5. Flexible vs. Rigid Spaces: Implications and Open Directions

The scope of flexibility:

• All known flexible examples are either induced by Hilbertian splitting at $p=2$, by degenerate or ultrametric intervals at $p=1$, or by deliberate product constructions supporting fiberwise exotic maps.
• Spaces of negative curvature, rays, cylinders over non-branching geodesic bases, and spherical suspensions of small diameter, are all proven rigid with respect to Wasserstein spaces ($p>1$) (Che et al., 2024, Bertrand et al., 2010, Bertrand et al., 2014).

Consequences:

• Flexibility can be "forced" universally: for every $p\geq1$ and any $X$, there exists an extended space $Y$ containing $X$ so that $W_p(Y)$ is flexible (Balogh et al., 13 Feb 2025).
• Conversely, any complete separable $X$ can be embedded into a $Y$ with a rigid $W_1(Y)$.

Open problems and conjectures:

• Whether a finite or countable augmentation of points always suffices to enforce rigidity in flexible cases remains open (Balogh et al., 26 Jan 2026).
• The classification of isometric rigidity vs. flexibility for other classes (e.g., Carnot groups for $p=1$, other fibered structures) is ongoing (Balogh et al., 2023).
• The geometric and group-theoretic mechanisms determining transition thresholds between flexibility and rigidity in parameter space or metric structure are not fully characterized.

6. Geometric Boundaries, Large-Scale Properties, and Rank

Flexible Wasserstein spaces, especially over non-Hilbert or non-Euclidean bases, display distinctive geometric features:

• In Hadamard spaces, the geodesic boundary $\partial W(X)$ is identified with the space of probability measures of unit speed on the Euclidean cone $c\partial X$, with cone-metric $d_\infty$; this induces a snowflake metric on the boundary and precludes isometric embeddings of Euclidean planes, enforcing rank one (Bertrand et al., 2010).
• Flexible spaces (e.g., $W_2(\mathbb{R}^n)$) have metric rank $n$: isometric copies of $\mathbb{R}^k$ exist exactly up to $k\leq n$. For Heisenberg groups $\mathbb{H}^n$, $W_p(\mathbb{H}^n)$ has rank $n$, and isometries are exclusively push-forwards, i.e., spaces are isometrically rigid (Balogh et al., 2023).

7. Broader Context in Metric Measure Geometry and Applied Directions

The understanding of isometric flexibility is central in both pure and applied settings:

• In metric measure geometry, flexibility characterizes the rare presence of higher-order symmetries and rich isometry group structure in Wasserstein spaces, contrasting with the rigidity anticipated from the underlying base geometry.
• For data science and geometric data analysis, rigidity assures that all Wasserstein distance-preserving embeddings of empirical measures arise from isometries of point clouds, supporting applications in shape matching and point-set correspondence (Balogh et al., 26 Jan 2026).
• Emergent connections to Banach space geometry, especially through the Arens–Eells construction and the universal $W_1$-normed space, provide a linear-algebraic framework to study Wasserstein metrics for persistence diagrams and signed measures (Bubenik et al., 2020).

A plausible implication is that the rare, highly structured appearance of isometric flexibility in Wasserstein spaces signals deep metric or measure-theoretic phenomena, often linked to Hilbertian structure, degenerate metrics, or specifically engineered product geometries. The sharp contrast between flexible and rigid cases underlines the sensitivity of Wasserstein geometry to both the base space and the transport cost exponent.