Otto-Wasserstein Geometry & Optimal Transport

• Otto-Wasserstein geometry is the study of Wasserstein spaces, where probability measures are endowed with metrics from optimal transport, emphasizing geodesic structure and curvature.
• It examines how geometric properties of Hadamard spaces, such as non-positive curvature and multiple geodesics, influence the structure of the associated measure spaces.
• The framework bridges optimal transport, boundary theory, and geometric group theory, offering insights into large-scale asymptotics, rank rigidity, and measure dynamics.

Otto-Wasserstein geometry is the study of the metric and geometric structure of Wasserstein spaces, which are spaces of probability measures endowed with a distance arising from optimal transport. In "A geometric study of Wasserstein spaces: Hadamard spaces" (1010.0590), the authors investigate the geometry of the Wasserstein space $\mathscr{W}_2(X)$ when the base space $X$ is a Hadamard space—meaning $X$ is complete, simply connected, locally compact, and has globally non-positive sectional curvature (CAT(0)). Their results reveal subtleties in how the geometric properties of $X$ transfer to its associated space of measures, enriching the Otto formalism. The study addresses curvature, geodesics, asymptotic geometry, rank, and rigidity phenomena, establishing deep links between optimal transport, boundary theory, and geometric group theory.

1. Definition and Foundational Structure

A Wasserstein space $\mathscr{W}_2(X)$ is defined as the set of Borel probability measures on a metric space $X$ with finite second moment: $$\mathscr{W}_2(X) = \left\{ \mu\ |\ \int_X d^2(x_0, x)\, d\mu(x) < \infty \text{ for some } x_0 \right\},$$ endowed with the 2-Wasserstein metric,

$$W_2^2(\mu_0, \mu_1) = \inf_{\gamma \in \Gamma(\mu_0, \mu_1)} \int_{X\times X} d^2(x, y) \, d\gamma(x, y),$$

where $\Gamma(\mu_0, \mu_1)$ is the set of couplings with given marginals.

When $X$ is a Hadamard space, $\mathscr{W}_2(X)$ inherits the property of being a geodesic space: any two measures can be connected by a constant-speed geodesic. However, $\mathscr{W}_2(X)$ (except for the case $X = \mathbb{R}$) does not retain the CAT(0) property, meaning unique geodesics and global non-positive curvature do not in general survive to Wasserstein spaces over higher-rank or more complex Hadamard spaces.

2. Curvature, Geodesics, and Non-Positive Curvature

While Hadamard spaces $X$ are geodesic and have non-positive curvature, the Wasserstein space $\mathscr{W}_2(X)$ only preserves the geodesic property, not the curvature. Specifically:

• If $X$ is a line ($\mathbb{R}$), $\mathscr{W}_2(X)$ is CAT(0).
• If $X$ is not a line, the abundance of branching geodesics in $X$ leads to an abundance of geodesics in $\mathscr{W}_2(X)$, causing the failure of unique geodesics and non-positive curvature.

Illustrative example: If $X$ has at least two geodesics between some points, measures of the form $\mu = \frac{1}{2}\delta_x + \frac{1}{2}\delta_{x'}$, $\nu = \frac{1}{2}\delta_y + \frac{1}{2}\delta_z$ can be connected by multiple optimal plans, giving rise to highly non-unique geodesics in $\mathscr{W}_2(X)$.

Even though CAT(0) fails for most $X$, $\mathscr{W}_2(X)$ retains large-scale geometry reminiscent of $X$, which is striking given the loss of local curvature properties.

3. Large-Scale and Asymptotic Geometry: Geodesic Boundary and the Cone Structure

A key contribution is the description of the geodesic boundary of $\mathscr{W}_2(X)$ and its relation to the boundary at infinity of $X$, denoted $\partial X$. For a Hadamard space, the geodesic boundary $\partial X$ is the set of asymptotic classes of geodesic rays.

For $\mathscr{W}_2(X)$, the analogous boundary is described as: $$\partial(\mathscr{W}_2(X)) = \mathscr{R}_1(\mathscr{W}_2(X))/\sim,$$ with rays corresponding to unit-speed geodesic rays.

Crucially, boundary points of Wasserstein space correspond to probability measures on the cone over the boundary of $X$: $$P_1(c\partial X) = \left\{ \mu \in P(c\partial X) : \int v^2 d\mu(v) = 1 \right\},$$ where the cone $c\partial X = (\partial X \times [0, \infty))/\sim$ is constructed by identifying $(\xi, 0)$ for all $\xi$.

Key asymptotic formula: For rays in $\mathscr{W}_2(X)$ (with asymptotic measures $\mu_\infty, \sigma_\infty$), the large-scale distance satisfies

$$\lim_{t\to\infty} \frac{W_2(\mu_t, \sigma_t)}{t} = W(\mu_\infty, \sigma_\infty),$$

where the right-hand side is the Wasserstein metric on the cone-over-boundary space with cost

$$d_\infty^2((\xi, s), (\xi', t)) = s^2 + t^2 - 2st \cos\angle(\xi, \xi').$$

Thus, the geometry "at infinity" of $\mathscr{W}_2(X)$ is governed by the Wasserstein geometry over the cone on the visual boundary of $X$.

4. Visibility, Rank, and Non-Embeddability

Visibility spaces—Hadamard spaces where any two points at infinity are joined by a geodesic—show strong rigidity in their Wasserstein geometries:

• Rank 1 rigidity: If $X$ is a visibility space (e.g., hyperbolic, or symmetric spaces with strictly negative curvature), then $\mathscr{W}_2(X)$ has geometric rank 1. Explicitly, there is no isometric embedding of the Euclidean plane $\mathbb{R}^2$ into $\mathscr{W}_2(X)$, because the metric on boundary measures “snowflakes” the total variation metric, preventing the existence of nontrivial rectifiable curves:

$W(\mu, \sigma) = 2 |\mu - \sigma|_{TV}^{1/2}$

Thus, Wasserstein spaces over negatively curved or visibility Hadamard spaces are highly rigid and “low-rank” compared to their Euclidean counterparts.

Contrast: For $X = \mathbb{R}^n$, $\mathscr{W}_2(X)$ can admit large-dimensional isometrically embedded Euclidean flats, and the boundary construction is much less singular (the boundary is a sphere with standard metric).

5. Homeomorphism, Boundary Topology, and Local Compactness

A homeomorphism is established between the geodesic boundary of Wasserstein space and the space of measures on the cone over the base’s boundary: $\partial(\mathscr{W}_2(X)) \simeq P_1(c\partial X)$ with respect to the cone topology.

• This gives a concrete, measure-theoretic model for the ideal boundary and for describing geodesic behavior at infinity.
• For noncompact $X$, $\mathscr{W}_2(X)$ is not locally compact; while its boundary at infinity encodes large-scale geometry, it does not compactify the Wasserstein space.

6. Summary Table: Key Properties

Feature	Hadamard Base $X$	Wasserstein $\mathscr{W}_2(X)$
Geodesic, CAT(0)	Yes	Geodesic, only CAT(0) if $X = \mathbb{R}$
Boundary at infinity	$\partial X$	$P_1(c\partial X)$ (measures on cone over boundary)
Large-scale metric	Angular/visual metric	Cone-over-boundary Wasserstein
Local compactness	Yes	Only if $X$ is compact
Rank	Usual geometric rank	At most 1 if $X$ is visibility space
$\mathbb{R}^2$ embedding	Possible in $X$ Euclidean	Not in visibility spaces

7. Significance and Broader Context

The Otto-Wasserstein geometry on Hadamard spaces provides a profound and precise connection between the shape of the base space and the analysis of its probability measures under optimal transport:

• Large-scale geometric features (asymptotics, boundary phenomena, rank) of $X$ are reflected (often in “coarse” yet precise ways) in $\mathscr{W}_2(X)$.
• The boundary of Wasserstein space is, in a deep sense, a space of measures on the cone over the boundary of $X$, echoing Otto’s philosophy that geometric and analytic properties of measure spaces are governed by the geometry of the base space at infinity.
• In strictly negatively curved (visibility) spaces, Wasserstein spaces are rigid with no isometric flat subspaces ($\mathbb{R}^2$), unlike the Euclidean baseline where high-dimensional flats exist.

This work bridges Otto’s formal calculus, modern metric geometry, and optimal transportation, yielding a robust framework for understanding the geometry, analysis, and group actions in spaces of measures over non-positively curved and especially negatively curved spaces.