An intrinsic parallel transport in Wasserstein space

TL;DR

This paper introduces a geometric method for parallel transporting tangent cones along geodesics in Wasserstein space over a Riemannian manifold, aligning with previous formal calculations in smooth cases.

Contribution

It provides a rigorous geometric construction of parallel transport in Wasserstein space, extending formal calculations to a precise mathematical framework.

Findings

Constructs a geometric parallel transport in Wasserstein space

Shows agreement with formal calculations in smooth cases

Extends understanding of geometric structures in optimal transport

Abstract

If M is a smooth compact connected Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent cones along geodesics in P(M). We show that when everything is smooth, the geometric parallel transport agrees with earlier formal calculations.

Equations

$$(\nabla {\eta }_Q,\nabla {\eta }_Q(0),\nabla {\eta }_Q(1))\in L^2([0,1];L^2(TM,dc(t)))\oplus L^2(TM,dc(0))\oplus L^2(TM,dc(1))$$

$$P^{\infty }(M)=\{\rho {\mathrm{dvol}}_M:\rho \in C^{\infty }(M),\rho >0,{\int }_M\rho {\mathrm{dvol}}_M=1\}.$$

$$(V_{\phi}F)(\rho\operatorname{dvol}_{M})\>=\>\frac{d}{d\epsilon}\Big{|}_{\epsilon=0}F\left(\rho\operatorname{dvol}_{M}\>-\>\epsilon\>\nabla^{i}(\rho\nabla_{i}\phi)\operatorname{dvol}_{M}\right).$$

$$⟨V_{{\varphi }_1},V_{{\varphi }_2}⟩(\rho {\mathrm{dvol}}_M)$$

$$=-{\int }_M{\varphi }_1{\nabla }^i(\rho {\nabla }_i{\varphi }_2){\mathrm{dvol}}_M.$$

$$⟨V_{{\varphi }_1},V_{{\varphi }_2}⟩(\rho {\mathrm{dvol}}_M)={\int }_M{\varphi }_1{\delta }_{V_{{\varphi }_2}}\rho {\mathrm{dvol}}_M={\int }_M{\varphi }_2{\delta }_{V_{{\varphi }_1}}\rho {\mathrm{dvol}}_M.$$

$$\frac{\partial \rho }{\partial t}+{\nabla }^j\left(\rho {\nabla }_j\varphi \right)=0.$$

$${\nabla }_i\left(\rho \left({\nabla }^i\frac{\partial \eta }{\partial t}+{\nabla }_j\varphi {\nabla }^i{\nabla }^j\eta \right)\right)=0.$$

$$\frac{d}{dt}{\int }_M⟨\nabla f,\nabla \eta ⟩d{\mu }_t={\int }_M⟨\nabla \frac{\partial f}{\partial t},\nabla \eta ⟩d{\mu }_t+{\int }_M{\mathrm{Hess}}_f(\nabla \eta ,\nabla \varphi )d{\mu }_t.$$

$$\frac{d}{dt}{\int }_M⟨\nabla f,\nabla \eta ⟩d{\mu }_t=$$

$$=$$

$${\int }_M⟨\nabla f,\nabla \eta ⟩\frac{\partial \rho }{\partial t}{\mathrm{dvol}}_M$$

$$\frac{d}{dt}{\int }_M⟨\nabla f,\nabla \eta ⟩d{\mu }_t-{\int }_M⟨\nabla \frac{\partial f}{\partial t},\nabla \eta ⟩d{\mu }_t$$

$$={\int }_M({\nabla }_{i}f)\left({\nabla }^i\frac{\partial \eta }{\partial t}\right)\rho {\mathrm{dvol}}_M-{\int }_M({\nabla }_{i}f)({\nabla }^i\eta ){\nabla }^j\left(\rho {\nabla }_j\varphi \right){\mathrm{dvol}}_M$$

$$=-{\int }_{M}f{\nabla }_i\left(\rho {\nabla }^i\frac{\partial \eta }{\partial t}\right){\mathrm{dvol}}_M-{\int }_M({\nabla }_{i}f)({\nabla }^i\eta ){\nabla }^j\left(\rho {\nabla }_j\varphi \right){\mathrm{dvol}}_M$$

$$={\int }_{M}f{\nabla }_i\left(\rho ({\nabla }_j\varphi )({\nabla }^i{\nabla }^j\eta )\right){\mathrm{dvol}}_M+{\int }_M{\nabla }^j(({\nabla }_{i}f)({\nabla }^i\eta ))({\nabla }_j\varphi )\rho {\mathrm{dvol}}_M$$

$$=-{\int }_M({\nabla }_{i}f)({\nabla }_j\varphi )({\nabla }^i{\nabla }^j\eta )\rho {\mathrm{dvol}}_M$$

$$+{\int }_M{\nabla }^j(({\nabla }_{i}f)({\nabla }^i\eta ))({\nabla }_j\varphi )\rho {\mathrm{dvol}}_M$$

$$={\int }_M({\nabla }^j{\nabla }_{i}f)({\nabla }^i\eta )({\nabla }_j\varphi )\rho {\mathrm{dvol}}_M$$

$$={\int }_M{\mathrm{Hess}}_f(\nabla \eta ,\nabla \varphi )d{\mu }_t.$$

$${\int }_M⟨\nabla f(1),\nabla {\eta }_1⟩d{\mu }_1-{\int }_M⟨\nabla f(0),\nabla {\eta }_0⟩d{\mu }_0=$$

$${\int }_0^1{\int }_M\left(⟨\nabla \frac{\partial f}{\partial t},\nabla \eta ⟩+{\mathrm{Hess}}_f(\nabla \eta ,\nabla \varphi )\right)d{\mu }_{t}dt$$

$${\int }_M⟨\nabla f(1),\nabla {\eta }_1⟩d{\mu }_1-{\int }_M⟨\nabla f(0),\nabla {\eta }_0⟩d{\mu }_0=$$

$${\int }_M⟨\nabla f(1),\nabla \eta (1)⟩d{\mu }_1-{\int }_M⟨\nabla f(0),\nabla \eta (0)⟩d{\mu }_0+$$

$${\int }_0^1{\int }_{M}f{\nabla }_i\left({\nabla }^i\frac{\partial \eta }{\partial t}+{\nabla }_j\varphi {\nabla }^i{\nabla }^j\eta \right)d{\mu }_{t}dt.$$

$${\int }_M|\nabla {\eta }_0{|}^{2}d{\mu }_0\le {\int }_M|\nabla {\eta }_1{|}^{2}d{\mu }_1$$

$${\int }_0^1{\int }_M|\nabla \eta {|}^{2}d{\mu }_{t}dt\le {\int }_M|\nabla {\eta }_1{|}^{2}d{\mu }_1$$

$${\int }_M⟨\nabla f(0),\nabla {\eta }_0⟩d{\mu }_0={\int }_M⟨\nabla f(1),\nabla {\eta }_1⟩d{\mu }_1={\int }_M⟨\nabla {\eta }_1,\nabla {\eta }_1⟩d{\mu }_1.$$

$${\int }_M⟨\nabla f(0),\nabla f(0)⟩d{\mu }_0={\int }_M⟨\nabla f(1),\nabla f(1)⟩d{\mu }_1={\int }_M⟨\nabla {\eta }_1,\nabla {\eta }_1⟩d{\mu }_1.$$

$${\int }_M|\nabla ({\eta }_0-f(0)){|}^{2}d{\mu }_0={\int }_M|\nabla {\eta }_0{|}^{2}d{\mu }_0-{\int }_M|\nabla {\eta }_1{|}^{2}d{\mu }_1\le 0.$$

$${\int }_0^1{\int }_M⟨\nabla f,\nabla \eta ⟩d{\mu }_{t}dt={\int }_M⟨\nabla f(1),\nabla {\eta }_1⟩d{\mu }_1={\int }_M⟨\nabla {\eta }_1,\nabla {\eta }_1⟩d{\mu }_1.$$

$${\int }_0^1{\int }_M|\nabla f-\nabla \eta {|}^{2}d{\mu }_{t}dt=$$

$${\int }_0^1{\int }_M|\nabla f{|}^{2}d{\mu }_{t}dt-2{\int }_0^1{\int }_M⟨\nabla f,\nabla \eta ⟩d{\mu }_{t}dt+{\int }_0^1{\int }_M|\nabla \eta {|}^{2}d{\mu }_{t}dt=$$

$${\int }_M|\nabla f(1){|}^{2}d{\mu }_1-2{\int }_M|\nabla {\eta }_1{|}^{2}d{\mu }_1+{\int }_0^1{\int }_M|\nabla \eta {|}^{2}d{\mu }_{t}dt=$$

$${\int }_0^1{\int }_M|\nabla \eta {|}^{2}d{\mu }_{t}dt-{\int }_M|\nabla {\eta }_1{|}^{2}d{\mu }_1\le 0.$$

$$P_0\circ P_1\circ \dots \circ P_{Q-1}={\left((d{\exp }_{{\gamma }_0(0)}{)}^{-1}\circ (d{\exp }_{{\gamma }_1(0)}{)}^{-1}\circ \dots \circ (d{\exp }_{{\gamma }_{Q-1}(0)}{)}^{-1}\right)}_{\ast }.$$

$$\underset{t\in [0,1]}{\sup }\Vert \varphi (t){\Vert }_{C^2(M)}<\infty .$$

$${\int }_{M}fd{c}_i(u)={\int }_{M}f(F_{i,u}(x))d{\mu }_{i,0}(x).$$

Full text

Research partially supported by NSF grant DMS-1207654 and a Simons Fellowship

1. Introduction

Let $M$ be a smooth compact connected Riemannian manifold without boundary. The space $P(M)$ of probability measures of $M$ carries a natural metric, the Wasserstein metric, and acquires the structure of a length space. There is a close relation between minimizing geodesics in $P(M)$ and optimal transport between measures. For more information on this relation, we refer to Villani’s book [13].

Otto discovered a formal Riemannian structure on $P(M)$, underlying the Wasserstein metric [10]. One can do formal geometric calculations for this Riemannian structure [6]. It is an interesting problem to make these formal considerations into rigorous results in metric geometry.

If $M$ has nonnegative sectional curvature then $P(M)$ is a compact length space with nonnegative curvature in the sense of Alexandrov [8, Theorem A.8], [12, Proposition 2.10]. Hence one can define the tangent cone $T_{\mu}P(M)$ of $P(M)$ at a measure $\mu\in P(M)$. If $\mu$ is absolutely continuous with respect to the volume form $\operatorname{dvol}_{M}$ then $T_{\mu}P(M)$ is a Hilbert space [8, Proposition A.33]. More generally, one can define tangent cones of $P(M)$ without any curvature assumption on $M$, using Ohta’s $2$-uniform structure on $P(M)$ [9]. Gigli showed that $T_{\mu}P(M)$ is a Hilbert space if and only if $\mu$ is a “regular” measure, meaning that it gives zero measure to any hypersurface which, locally, is the graph of the difference of two convex functions [3, Corollary 6.6]. For examples of tangent cones at nonregular measures, if $S$ is an embedded submanifold of $M$, and $\mu$ is an absolutely continuous measure on $S$, then $T_{\mu}P(M)$ was computed in [7, Theorem 1.1].

If $\gamma\>:\>[0,1]\rightarrow M$ is a smooth curve in a Riemannian manifold then one can define the (reverse) parallel transport along $\gamma$ as a linear isometry from $T_{\gamma(1)}M$ to $T_{\gamma(0)}M$. If $X$ is a finite-dimensional Alexandrov space then the replacement of a tangent space is a tangent cone. If one wants to define a parallel transport along a curve $c:[0,1]\rightarrow X$, as a map from $T_{c(1)}X$ to $T_{c(0)}X$, then there is the problem that the tangent cones along $c$ may not look much alike. For example, the curve $c$ may pass through various strata of $X$. One can deal with this problem by assuming that $c$ is in the interior of a minimizing geodesic. In this case, Petrunin proved the tangent cones along $c$ are mutually isometric, by constructing a parallel transport map [11]. His construction of the parallel transport map was based on passing to a subsequential limit in an iterative construction along $c$. It is not known whether the ensuing parallel transport is uniquely defined, although this is irrelevant for Petrunin’s result.

In the case of a smooth curve $c\>:\>[0,1]\rightarrow P^{\infty}(M)$ in the space of smooth probability measures, one can do formal Riemannian geometry calculations on $P^{\infty}(M)$ to write down an equation for parallel transport along $c$ [6, Proposition 3]. It is a partial differential equation in terms of a family of functions $\{\eta_{t}\}_{t\in[0,1]}$. Ambrosio and Gigli noted that there is a weak version of this partial differential equation [1, (5.9)]. By a slight extension, we will define weak solutions to the formal parallel transport equation; see Definition 2.13.

Petrunin’s construction of parallel transport cannot work in full generality on $P(M)$, since Juillet showed that there is a minimizing Wasserstein geodesic $c$ with the property that the tangent cones at measures on the interior of $c$ are not all mutually isometric [5]. However one can consider applying the construction on certain convex subsets of $P(M)$. We illustrate this in two cases. The first and easier case is when $c$ is a Wasserstein geodesic of $\delta$-measures (Proposition 3.1). The second case is when $c$ is a Wasserstein geodesic of absolutely continuous measures, lying in the interior of a minimizing Wasserstein geodesic, and satisfying a regularity condition. Suppose that $\nabla\eta_{1}\in T_{c(1)}P(M)$ is an element of the tangent cone at the endpoint. Here $\nabla\eta_{1}\in L^{2}(TM,dc(1))$ is a square-integrable gradient vector field on $M$ and $\eta_{1}$ is in the Sobolev space $H^{1}(M,dc(1))$. For each sufficiently large integer $Q$, we construct a triple

[TABLE]

with $\nabla\eta_{Q}(1)=\nabla\eta_{1}$, which represents an approximate parallel transport along $c$.

Theorem 1.2**.**

Suppose that $M$ has nonnegative sectional curvature. A subsequence of $\{(\nabla\eta_{Q},\nabla\eta_{Q}(0),\nabla\eta_{Q}(1))\}_{Q=1}^{\infty}$ converges weakly to a weak solution $(\nabla\eta_{\infty},\nabla\eta_{\infty,0},\nabla\eta_{\infty,1})$ of the parallel transport equation with $\nabla\eta_{\infty,1}=\nabla\eta_{1}$. If $c$ is a smooth geodesic in $P^{\infty}(M)$, $\eta_{1}$ is smooth, and there is a smooth solution $\eta$ to the parallel transport equation (2.6) with $\eta(1)=\eta_{1}$, then $\lim_{Q\rightarrow\infty}(\nabla\eta_{Q},\nabla\eta_{Q}(0),\nabla\eta_{Q}(1))=(\nabla\eta,\nabla\eta(0),\nabla\eta(1))$ in norm.

Remark 1.3**.**

In the setting of Theorem 1.2, we can say that $\nabla\eta_{\infty,0}$ is the parallel transport of $\nabla\eta_{1}$ along $c$ to $T_{c(0)}P(M)$.

Remark 1.4**.**

We are assuming that $M$ has nonnegative sectional curvature in order to apply some geometric results from [11]. It is likely that this assumption could be removed.

Remark 1.5**.**

A result related to Theorem 1.2 was proven by Ambrosio and Gigli when $M={\mathbb{R}}^{n}$ [1, Theorem 5.14], and extended to general $M$ by Gigli [4, Theorem 4.9]. As explained in [1, 4], the construction of parallel transport there can be considered to be extrinsic, in that it is based on embedding the (linear) tangent cones into a Hilbert space and applying projection operators to form the approximate parallel transports. Although we instead use Petrunin’s intrinsic construction, there are some similarities between the two constructions; see Remark 3.32. We use some techniques from [1], especially the idea of a weak solution to the parallel transport equation.

Remark 1.6**.**

Besides its inherent naturality, the intrinsic construction of parallel transport given here is likely to allow for extensions. For example, using the results of [7], it seems likely that Petrunin’s construction could be extended to define parallel transport along Wasserstein geodesics of absolutely continuous measures on submanifolds of $M$. In the present paper we have done this when the submanifolds have dimension zero or codimension zero.

The structure of this paper is as follows. In Section 2 we discuss weak solutions to the parallel transport equation. In Section 3 we prove Theorem 1.2.

I thank Takumi Yokota and Nicola Gigli for references to the literature.

2. Weak solutions to the parallel transport equation

Let $M$ be a compact connected Riemannian manifold without boundary. Put

[TABLE]

Given $\phi\in C^{\infty}(M)$, define a vector field $V_{\phi}$ on $P^{\infty}(M)$ by saying that for $F\in C^{\infty}(P^{\infty}(M))$,

[TABLE]

The map $\phi\rightarrow V_{\phi}$ passes to an isomorphism $C^{\infty}(M)/{\mathbb{R}}\rightarrow T_{\rho\operatorname{dvol}_{M}}P^{\infty}(M)$. Otto’s Riemannian metric on $P^{\infty}(M)$ is given [10] by

[TABLE]

In view of (2.2), we write $\delta_{V_{\phi}}\rho\>=\>-\>\nabla^{i}(\rho\nabla_{i}\phi)$. Then

[TABLE]

To write the equation for parallel transport, let $c\>:\>[0,1]\rightarrow P^{\infty}(M)$ be a smooth curve. We write $c(t)\>=\>\mu_{t}\>=\>\rho(t)\>\operatorname{dvol}_{M}$ and define $\phi(t)\in C^{\infty}(M)$, up to a constant, by $\frac{dc}{dt}\>=\>V_{\phi(t)}$. This is the same as saying

[TABLE]

Let $V_{\eta(t)}$ be a vector field along $c$, with $\eta(t)\in C^{\infty}(M)$. The equation for $V_{\eta}$ to be parallel along $c$ [6, Proposition 3] is

[TABLE]

Lemma 2.7**.**

[6, Lemma 5]** If $\eta,\overline{\eta}$ are solutions of (2.6) then $\int_{M}\langle\nabla\eta,\nabla\overline{\eta}\rangle\>d\mu_{t}$ is constant in $t$.

Lemma 2.8**.**

Given $\eta_{1}\in C^{\infty}(M)$, there is at most one solution of (2.6) with $\eta(1)=\eta_{1}$, up to time-dependent additive constants.

Proof.

By linearity, it suffices to consider the case when $\eta_{1}=0$. From Lemma 2.7, $\nabla\eta(t)=0$ and so $\eta(t)$ is spatially constant. ∎

For consistency with later notation, we will write $C^{\infty}([0,1];C^{\infty}(M))$ for $C^{\infty}([0,1]\times M)$.

Lemma 2.9**.**

(c.f. [1, (5.8)]) Given $f\in C^{\infty}([0,1];C^{\infty}(M))$, if $\eta$ satisfies (2.6) then

[TABLE]

Proof.

We have

[TABLE]

Then

[TABLE]

This proves the lemma. ∎

We now weaken the regularity assumptions. Let $P^{ac}(M)$ denote the absolutely continuous probability measures on $M$ with full support. Suppose that $c\>:\>[0,1]\rightarrow P^{ac}(M)$ is a Lipschitz curve whose derivative $c^{\prime}(t)\in T_{c(t)}P(M)$ exists for almost all $t$. We can write $c^{\prime}(t)=V_{\phi(t)}$ with $\nabla\phi(t)\in L^{2}(TM,dc(t))$. By the Lipschitz assumption, the essential supremum over $t\in[0,1]$ of $\|\nabla\phi(t)\|_{L^{2}(TM,dc(t))}$ is finite. As before, we write $c(t)=\mu_{t}$.

Definition 2.13**.**

Let $c\>:\>[0,1]\rightarrow P^{ac}(M)$ be a Lipschitz curve whose derivative $c^{\prime}(t)\in T_{c(t)}P(M)$ exists for almost all $t$. Given $\nabla\eta_{0}\in L^{2}(TM,d\mu_{0})$, $\nabla\eta_{1}\in L^{2}(TM,d\mu_{1})$ and $\nabla\eta\in L^{2}([0,1];L^{2}(TM,d\mu_{t}))$, we say that $(\nabla\eta,\nabla\eta_{0},\nabla\eta_{1})$ is a weak solution of the parallel transport equation if

[TABLE]

for all $f\in C^{\infty}([0,1];C^{\infty}(M))$.

Remark 2.15**.**

In what follows, there would be analogous results if we replaced $C^{\infty}([0,1];C^{\infty}(M))$ everywhere by $C^{0}([0,1];C^{2}(M))\cap C^{1}([0,1];C^{1}(M))$. We will stick with $C^{\infty}([0,1];C^{\infty}(M))$ for concreteness.

From Lemma 2.9, if $c$ is a smooth curve in $P^{\infty}(M)$ and $\eta\in C^{\infty}([0,1];C^{\infty}(M))$ is a solution of (2.6) then $(\nabla\eta,\nabla\eta(0),\nabla\eta(1))$ is a weak solution of the parallel transport equation. We now prove the converse.

Lemma 2.16**.**

Suppose that $c$ is a smooth curve in $P^{\infty}(M)$. Given $\eta_{0},\eta_{1}\in C^{\infty}(M)$ and $\eta\in C^{\infty}([0,1];C^{\infty}(M))$, if $(\nabla\eta,\nabla\eta_{0},\nabla\eta_{1})$ is a weak solution of the parallel transport equation then $\eta$ satisfies (2.6), $\eta(0)=\eta_{0}$ and $\eta(1)=\eta_{1}$ (modulo constants).

Proof.

In this case, equation (2.14) is equivalent to

[TABLE]

Taking $f\in C^{\infty}([0,1];C^{\infty}(M))$ with $f(0)=f(1)=0$, it follows that (2.6) must hold. Then taking all $f\in C^{\infty}([0,1];C^{\infty}(M))$, it follows that $\nabla\eta_{0}=\nabla\eta(0)$ and $\nabla\eta_{1}=\nabla\eta(1)$. Hence $\eta(0)=\eta_{0}$ and $\eta(1)=\eta_{1}$ (modulo constants). ∎

Lemma 2.18**.**

Suppose that $c$ is a smooth curve in $P^{\infty}(M)$. Given $\nabla\eta_{0}\in L^{2}(TM,d\mu_{0})$, $\nabla\eta_{1}\in L^{2}(TM,d\mu_{1})$, $\nabla\eta\in L^{2}([0,1];L^{2}(TM,d\mu_{t}))$ and $f\in C^{\infty}([0,1];C^{\infty}(M))$, suppose that

1. (1)

$(\nabla\eta,\nabla\eta_{0},\nabla\eta_{1})$* is a weak solution to the parallel transport equation,* 2. (2)

$f$* satisfies (2.6),* 3. (3)

$\nabla f(1)=\nabla\eta_{1}$, 4. (4)

[TABLE]

and 5. (5)

[TABLE]

Then $\nabla f(0)=\nabla\eta_{0}$, and $\nabla f(t)=\nabla\eta(t)$ for almost all $t$.

Proof.

From (2.6) (applied to $f$) and (2.14), we have

[TABLE]

From Lemma 2.7,

[TABLE]

Then

[TABLE]

Thus $\nabla f(0)=\nabla\eta_{0}$ in $L^{2}(TM,d\mu_{0})$.

Next, replacing $f$ by $tf$ in (2.14) gives

[TABLE]

Then

[TABLE]

Thus $\nabla f(t)=\nabla\eta(t)$ in $L^{2}(TM,d\mu_{t})$, for almost all $t$. ∎

3. Parallel transport along Wasserstein geodesics

3.1. Parallel transport in a finite-dimensional Alexandrov space

We recall the construction of parallel transport in a finite-dimensional Alexandrov space $X$.

Let $c:[0,1]\rightarrow X$ be a geodesic segment that lies in the interior of a minimizing geodesic. Then $T_{c(t)}X$ is an isometric product of ${\mathbb{R}}$ with the normal cone $N_{c(t)}X$. We want to construct a parallel transport map from $N_{c(1)}X$ to $N_{c(0)}X$.

Given $Q\in{\mathbb{Z}}^{+}$ and $0\leq i\leq Q-1$, define ${c}_{i}:[0,1]\rightarrow X$ by ${c}_{i}(u)=c\left(\frac{i+u}{Q}\right)$. We define an approximate parallel transport $P_{i}:N_{{c}_{i}(1)}X\rightarrow N_{{c}_{i}(0)}X$ as follows. Given $v\in N_{{c}_{i}(1)}X$, let $\gamma:[0,\epsilon]\rightarrow X$ be a minimizing geodesic segment with $\gamma(0)={c}_{i}(1)$ and $\gamma^{\prime}(0)=v$. For each $s\in(0,\epsilon]$, let $\mu_{s}:[0,1]\rightarrow X$ be a minimizing geodesic with $\mu_{s}(0)={c}_{i}(0)$ and $\mu_{s}(1)=\gamma(s)$. Let $w_{s}\in N_{{c}_{i}(0)}X$ be the normal projection of $\frac{1}{s}\mu_{s}^{\prime}(0)\in T_{{c}_{i}(0)}X$. After passing to a sequence $s_{i}\rightarrow 0$, we can assume that $\lim_{i\rightarrow\infty}w_{s_{i}}=w\in N_{{c}_{i}(0)}X$. Then $P_{i}(v)=w$. If $X$ has nonnegative Alexandrov curvature then $|w|\geq|v|$.

In [11], the approximate parallel transport from an appropriate dense subset $L_{Q}\subset N_{c(1)}X$ to $N_{c(0)}X$ was defined to be $P_{0}\circ P_{1}\circ\ldots\circ P_{Q-1}$. It was shown that by taking $Q\rightarrow\infty$ and applying a diagonal argument, in the limit one obtains an isometry from a dense subset of $N_{c(1)}X$ to $N_{c(0)}X$. This extends by continuity to an isometry from $N_{c(1)}X$ to $N_{c(0)}X$.

If $X$ is a smooth Riemannian manifold then $P_{i}$ is independent of the choices and can be described as follows. Given $v\in N_{{c}_{i}(1)}X$, let $j_{v}(u)$ be the Jacobi field along $c$ with $j_{v}(0)=0$ and $j_{v}(1)=v$. (It is unique since $c$ is in the interior of a minimizing geodesic.) Then $P_{i}(v)=j_{v}^{\prime}(0)$.

3.2. Construction of parallel transport along a

Wasserstein geodesic of delta measures

Let $M$ be a compact connected Riemannian manifold without boundary. Let $\gamma:[0,1]\rightarrow M$ be a geodesic segment that lies in the interior of a minimizing geodesic. Let $\Pi:T_{\gamma(1)}M\rightarrow T_{\gamma(0)}M$ be (reverse) parallel transport along $\gamma$. Put $c(t)=\delta_{\gamma(t)}\in P(M)$. Then $\{c(t)\}_{t\in[0,1]}$ is a Wasserstein geodesic that lies in the interior of a minimizing geodesic. We apply Petrunin’s construction to define parallel transport directly from the tangent cone $T_{c(1)}P(M)$ to the tangent cone $T_{c(0)}P(M)$ (instead of the normal cones). From [7, Theorem 1.1], we know that $T_{c(t)}P(M)\cong P_{2}(T_{\gamma(t)}M)$.

Proposition 3.1**.**

The parallel transport map from $T_{c(1)}P(M)\cong P_{2}(T_{\gamma(1)}M)$ to $T_{c(0)}P(M)\cong P_{2}(T_{\gamma(0)}M)$ is the map $\mu\rightarrow\Pi_{*}\mu$.

Proof.

Given $Q\in{\mathbb{Z}}^{+}$ and $0\leq i\leq Q-1$, define $\gamma_{i}:[0,1]\rightarrow M$ by $\gamma_{i}(u)=\gamma\left(\frac{i+u}{Q}\right)$ and ${c}_{i}:[0,1]\rightarrow P(M)$ by ${c}_{i}(u)=\delta_{\gamma_{i}(u)}$. We define an approximate parallel transport $P_{i}:T_{{c}_{i}(1)}P(M)\rightarrow T_{{c}_{i}(0)}P(M)$ as follows.

Given $s\in{\mathbb{R}}^{+}$ and a real vector space $V$, let $R_{s}:V\rightarrow V$ be multiplication by $s$. Let $\nu$ be a compactly-supported element of $P(T_{\gamma_{i}(1)}M)$. For small $\epsilon>0$, there is a Wasserstein geodesic $\sigma\>:\>{[0,\epsilon]}\rightarrow P(M)$, with $\sigma(0)=c_{i}(1)$ and $\sigma^{\prime}(0)$ corresponding to $\nu\in T_{c_{i}(1)}PM$, given by $\sigma(s)=(\exp_{\gamma_{i}(1)}\circ R_{s})_{*}\nu$. Given $s\in(0,\epsilon]$, let $\mu_{s}:[0,1]\rightarrow P(M)$ be a minimizing geodesic with $\mu_{s}(0)=c_{i}(0)=\delta_{\gamma_{i}(0)}$ and $\mu_{s}(1)=\sigma(s)$. There is a compactly-supported measure $\tau_{s}\in P_{2}(T_{\gamma_{i}(0)}M)=T_{c_{i}(0)}P(M)$ so that for $v\in[0,1]$, we have $\mu_{s}(v)=(\exp_{\gamma_{i}(0)}\circ R_{v})_{*}\tau_{s}$. If $Q$ is large and $\epsilon$ is small then all of the constructions take place well inside a totally convex ball, so $\tau_{s}$ is unique and can be written as $\tau_{s}=\left(\exp_{\gamma_{i}(0)}^{-1}\circ\exp_{\gamma_{i}(1)}\circ R_{s}\right)_{*}\nu$. Then $\lim_{s\rightarrow 0}\frac{1}{s}(\tau_{s}-\tau_{0})$ exists and equals $(d\exp_{\gamma_{i}(0)})^{-1}_{*}\nu$. Thus $P_{i}=(d\exp_{\gamma_{i}(0)})^{-1}_{*}$.

Now

[TABLE]

Taking $Q\rightarrow\infty$, this approaches $\Pi_{*}$. ∎

3.3. Construction of parallel transport along a

Wasserstein geodesic of absolutely continuous measures

Let $M$ be a compact connected boundaryless Riemannian manifold with nonnegative sectional curvature. Then $(P(M),W_{2})$ has nonnegative Alexandrov curvature.

Let $c:[0,1]\rightarrow P^{ac}(M)$ be a geodesic segment that lies in the interior of a minimizing geodesic. Write $c^{\prime}(t)=V_{\phi(t)}$. Since $\phi(t)$ is defined up to a constant, it will be convenient to normalize it by $\int_{M}\phi(t)\>d\mu_{t}=0$. We assume that

[TABLE]

In particular, this is satisfied if $c$ lies in $P^{\infty}(M)$.

Let $N_{c(t)}P(M)$ denote the normal cone to $c$ at $c(t)$. We want to construct a parallel transport map from $N_{c(1)}P(M)$ to $N_{c(0)}P(M)$.

Given $Q\in{\mathbb{Z}}^{+}$ and $0\leq i\leq Q-1$, define ${c}_{i}:[0,1]\rightarrow P(M)$ by ${c}_{i}(u)=c\left(\frac{i+u}{Q}\right)$. Correspondingly, write ${\mu}_{i,u}=\mu_{\frac{i+u}{Q}}$. We define an approximate parallel transport $P_{i}:N_{{c}_{i}(1)}P(M)\rightarrow N_{{c}_{i}(0)}P(M)$, using Jacobi fields, as follows.

Let us write ${c}_{i}^{\prime}(u)=V_{\phi_{i}(u)}$, i.e. $\phi_{i}(u)=\frac{1}{Q}\phi\left(\frac{i+u}{Q}\right)$. The curve ${c}_{i}$ is given by ${c}_{i}(u)=(F_{i,u})_{*}{c}_{i}(0)$, where $F_{i,u}(x)=\exp_{x}(u\nabla_{x}\phi_{i}(0))$. That is, for any $f\in C^{\infty}(M)$,

[TABLE]

If $\sigma_{i}$ is a variation of $\phi_{i}(0)$, i.e. $\delta\phi_{i}(0)=\sigma_{i}$, then taking the variation of (3.4) gives

[TABLE]

Here

[TABLE]

with $y=F_{i,u}(x)$. The corresponding tangent vector at ${c}_{i}(u)$ is represented by $L_{\sigma_{i}}(u)=\Pi_{{c}_{i}(u)}W_{\sigma_{i}}(u)$, where $\Pi_{{c}_{i}(u)}$ is orthogonal projection on $\overline{\operatorname{Im}\nabla}\subset L^{2}(TM,d\mu_{i,u})$. We can think of $J_{\sigma_{i}}(u)=uL_{\sigma_{i}}(u)$ as a Jacobi field along ${c}_{i}$. If $v=J_{\sigma_{i}}(1)=L_{\sigma_{i}}(1)=\Pi_{{c}_{i}(1)}W_{\sigma_{i}}(1)$ then its approximate parallel transport along ${c}_{i}$ is represented by $w=J_{\sigma_{i}}^{\prime}(0)=L_{\sigma_{i}}(0)=\nabla\sigma_{i}\in\overline{\operatorname{Im}\nabla}\subset L^{2}(TM,d\mu_{i,0})$.

Next, using (3.6), for $f\in C^{\infty}(M)$ we have

[TABLE]

Here $\partial_{u}$ is the vector at $F_{i,u}(x)$ given by

[TABLE]

If instead $f\in C^{\infty}([0,1];C^{\infty}(M))$ then

[TABLE]

We will need to estimate $\int_{M}|W_{\sigma_{i}}(u)-L_{\sigma_{i}}(u)|^{2}\>d{\mu}_{i,u}$.

Lemma 3.10**.**

For large $Q$, there is an estimate

[TABLE]

Here, and hereafter, $\operatorname{const.}$ denotes a constant that can depend on the fixed Riemannian manifold $(M,g)$.

Proof.

Since $\Pi_{c_{i}(u)}$ is projection onto $\overline{\operatorname{Im}(\nabla)}\subset L^{2}(TM,d\mu_{i,u})$, and $\nabla(\sigma_{i}\circ F_{i,u}^{-1})\in\operatorname{Im}(\nabla)$, we have

[TABLE]

(Compare with [1, Proposition 4.3].) Defining $T_{i,t,x}\>:\>T_{x}M\rightarrow T_{x}M$ by

[TABLE]

we obtain

[TABLE]

Since $\sup_{t\in[0,1]}\|\nabla\phi(t)\|_{C^{0}(M)}<\infty$, if $Q$ is large then $\|\nabla\phi_{i}(0)\|_{C^{0}(M)}$ is much smaller than the injectivity radius of $M$. In particular, the curve $\{F_{i,u}(x)\}_{u\in[0,1]}$ lies well within a normal ball around $x$. Now $T_{i,t,x}$ can be estimated in terms of $\operatorname{Hess}(\phi_{i})$. In general, if a function $h$ on a complete Riemannian manifold satisfies $\operatorname{Hess}(h)=0$ then the manifold isometrically splits off an ${\mathbb{R}}$-factor and the optimal transport path generated by $\nabla h$ is translation along the ${\mathbb{R}}$-factor. In such a case, the analog of $T_{i,t,x}$ is the identity map. If $\operatorname{Hess}(h)\neq 0$ then the divergence of a short optimal transport path from being a translation can be estimated in terms of $\operatorname{Hess}(h)$. Putting in the estimates gives (3.11). ∎

Using Lemma 3.10, we have

[TABLE]

Next, given $x\in M$, consider the geodesic

[TABLE]

Put

[TABLE]

Then $j_{\sigma_{i},x}$ is a Jacobi field along $\gamma_{i,x}$, with $j_{\sigma_{i},x}(0)=0$ and $j_{\sigma_{i},x}^{\prime}(0)=\nabla_{x}\sigma_{i}$. Jacobi field estimates give

[TABLE]

again for $Q$ large.

Lemma 3.19**.**

Define $A_{i}:\left(\overline{\operatorname{Im}(\nabla)}\subset L^{2}(TM,d\mu_{i,0})\right)\rightarrow\left(\overline{\operatorname{Im}(\nabla)}\subset L^{2}(TM,d\mu_{i,1})\right)$ by

[TABLE]

Then for large $Q$, the map $A_{i}$ is invertible for all $i\in\{0,\ldots,Q-1\}$.

Proof.

Define $B_{i}:\left(\overline{\operatorname{Im}(\nabla)}\subset L^{2}(TM,d\mu_{i,1})\right)\rightarrow\left(\overline{\operatorname{Im}(\nabla)}\subset L^{2}(TM,d\mu_{i,0})\right)$ by

[TABLE]

Then whenever $\nabla f\in L^{2}(TM,d\mu_{i,1})$, we have

[TABLE]

so whenever $\nabla f^{\prime}\in L^{2}(TM,d\mu_{i,1})$, for large $Q$ we have

[TABLE]

Hence $\|A_{i}B_{i}-I\|=o(Q)$, so for large $Q$ the map $A_{i}B_{i}$ is invertible and a right inverse for $A_{i}$ is given by $B_{i}(A_{i}B_{i})^{-1}$. This implies that $A_{i}$ is surjective.

Now suppose that $\nabla\sigma\in\operatorname{Ker}(A_{i})$ is nonzero, with $\sigma\in H^{1}(M,d\mu_{i,0})$. After normalizing, we may assume that $\nabla\sigma$ has unit length. Then

[TABLE]

for large $Q$. If $Q$ is sufficiently large then this is a contradiction, so $A_{i}$ is injective. ∎

Fix ${\mathcal{V}}_{1}\in N_{c(1)}P(M)$. If ${\mathcal{V}}_{1}\neq 0$ then after normalizing, we may assume that it has unit length. For $Q\in{\mathbb{Z}}^{+}$ large and $t\in[0,1]$, define ${\mathcal{V}}_{Q}(t)\in N_{c(t)}P(M)$ as follows. First, using Lemma 3.19, find $\sigma_{Q-1}$ so that ${\mathcal{V}}_{1}=L_{\sigma_{Q-1}}(1)$. For $t\in\left[\frac{Q-1}{Q},1\right]$, put

[TABLE]

Doing backward recursion, starting with $i=Q-2$, using Lemma 3.19 we find $\sigma_{i}$ so that $L_{\sigma_{i}}(1)=L_{\sigma_{i+1}}(0)=\nabla\sigma_{i+1}$. For $t\in\left[\frac{i}{Q},\frac{i+1}{Q}\right]$, put

[TABLE]

Decrease $i$ by one and repeat. The last step is when $i=0$.

From the argument in [11, Lemma 1.8],

[TABLE]

We note that the proof of [11, Lemma 1.8] only uses results about geodesics in Alexandrov spaces, it so applies to our infinite-dimensional setting. It also uses the assumption that $c$ lies in the interior of a minimizing geodesic. After passing to a subsequence, we can assume that

[TABLE]

in the weak topology on $L^{2}([0,1];L^{2}(TM,d\mu_{t}))\oplus L^{2}(TM,d\mu_{0})\oplus L^{2}(TM,d\mu_{1})$. Note that ${\mathcal{V}}_{\infty,1}={\mathcal{V}}_{1}$.

From (3.9), (3.15) and (3.18), for a fixed $f\in C^{\infty}([0,1];C^{\infty}(M))$, on each interval $\left[\frac{i}{Q},\frac{i+1}{Q}\right]$ we have

[TABLE]

It follows that $\left({\mathcal{V}}_{\infty},{\mathcal{V}}_{\infty,0},{\mathcal{V}}_{\infty,1}\right)$ is a weak solution of the parallel transport equation. As the limiting vector fields are gradient vector fields, we can write $\left({\mathcal{V}}_{\infty},{\mathcal{V}}_{\infty,0},{\mathcal{V}}_{\infty,1}\right)=\left(\nabla\eta_{\infty},\nabla\eta_{\infty,0},\nabla\eta_{\infty,1}\right)$ for some $\left(\eta_{\infty},\eta_{\infty,0},\eta_{\infty,1}\right)\in L^{2}([0,1];H^{1}(M,d\mu_{t}))\oplus H^{1}(M,d\mu_{0})\oplus H^{1}(M,d\mu_{1}))$.

Suppose that $c$ is a smooth geodesic in $P^{\infty}(M)$, that ${\mathcal{V}}_{1}$ (and hence $\eta_{\infty,1}$) is smooth and that there is a smooth solution $\eta$ to the parallel transport equation (2.6) with $\nabla\eta(1)=\nabla\eta_{\infty,1}$. By Lemma 2.7, $\|\nabla\eta(t)\|$ is independent of $t$. By Lemma 2.18, $\left(\nabla\eta_{\infty},\nabla\eta_{\infty,0},\nabla\eta_{\infty,1}\right)=\left(\nabla\eta,\nabla\eta(0),\nabla\eta(1)\right)$. We claim that

[TABLE]

in the norm topology on $L^{2}([0,1];L^{2}(TM,d\mu_{t}))\oplus L^{2}(TM,d\mu_{0})\oplus L^{2}(TM,d\mu_{1})$. This is because of the general fact that if $\{x_{i}\}_{i=1}^{\infty}$ is a sequence in a Hilbert space $H$ with $\lim_{i\rightarrow\infty}|x_{i}|=1$, and there is some unit vector $x_{\infty}\in H$ so that every weakly convergent subsequence of $\{x_{i}\}_{i=1}^{\infty}$ has weak limit $x_{\infty}$, then $\lim_{i\rightarrow\infty}x_{i}=x_{\infty}$ in the norm topology.

In particular,

[TABLE]

in the norm topology on $L^{2}(TM,d\mu_{0})$.

This proves Theorem 1.2.

Remark 3.32**.**

The construction of parallel transport in [1, Section 5] and [4, Section 4] is also by taking the limit of an iterative procedure. The underlying logic in [1, 4] is different than what we use, which results in a different algorithm. The iterative construction in [1, 4] amounts to going forward along the curve $c$ applying certain maps ${\mathcal{P}}_{i}$, instead of going backward along $c$ using the inverses of the $A_{i}$’s as we do. In the case of ${\mathbb{R}}^{n}$, the map ${\mathcal{P}}_{i}$ is the same as $A_{i}$, but this is not the case in general. The map ${\mathcal{P}}_{i}$ is nonexpanding, which helps the construction in [1, 4]. In contrast, $A_{i}^{-1}$ is not nonexpanding. In order to control its products, we use the result (3.27) from [11].