Let $(M,g)$ be a complete connected smooth Riemannian manifold. Let $d_g:M\times M\to[0,\infty)$ denote the Riemannian distance, let $\operatorname{vol}_g$ denote the Riemannian volume measure on $(M,\mathcal B(M))$, and let $\mathcal P_2(M)$ denote the set of Borel probability measures $\mu$ on $M$ satisfying

\begin{align*} \int_M d_g(x,x_0)^2\,d\mu(x)<\infty \end{align*}

for some, equivalently every, $x_0\in M$. Equip $\mathcal P_2(M)$ with the quadratic Wasserstein distance $W_2$ induced by $d_g$.

Let $\operatorname{Geo}(M)$ denote the Borel subset of $C([0,1];M)$, equipped with the compact-open topology, consisting of constant-speed minimizing geodesics $\gamma:[0,1]\to M$. For each $t\in[0,1]$, let $e_t:\operatorname{Geo}(M)\to M$ be the evaluation map $e_t(\gamma)=\gamma(t)$.

For $\mu\in\mathcal P_2(M)$, define the relative entropy with respect to $\operatorname{vol}_g$ by

\begin{align*} \operatorname{Ent}_{\operatorname{vol}_g}(\mu):=\int_M \rho\log\rho\,d\operatorname{vol}_g(x) \end{align*}

if $\mu=\rho\,\operatorname{vol}_g$ for a Borel density $\rho:M\to[0,\infty)$ and $\rho\log\rho\in L^1(M,\mathcal B(M),\operatorname{vol}_g)$, and set $\operatorname{Ent}_{\operatorname{vol}_g}(\mu)=+\infty$ otherwise.

Assume that there exists $K\in\mathbb R$ such that

\begin{align*} \operatorname{Ric}_p(v,v)\ge K g_p(v,v) \end{align*}

for every $p\in M$ and every $v\in T_pM$. Then, for every $\mu_0,\mu_1\in\mathcal P_2(M)$ with $\operatorname{Ent}_{\operatorname{vol}_g}(\mu_0),\operatorname{Ent}_{\operatorname{vol}_g}(\mu_1)\in\mathbb R$, there exist a Borel probability measure $\Pi$ on $\operatorname{Geo}(M)$ and a curve $\mu:[0,1]\to\mathcal P_2(M)$, $t\mapsto\mu_t$, such that $(e_t)_{\#}\Pi=\mu_t$ for every $t\in[0,1]$, the coupling $(e_0,e_1)_{\#}\Pi$ is $W_2$-optimal between $\mu_0$ and $\mu_1$, and

\begin{align*} \operatorname{Ent}_{\operatorname{vol}_g}(\mu_t)\le (1-t)\operatorname{Ent}_{\operatorname{vol}_g}(\mu_0)+t\operatorname{Ent}_{\operatorname{vol}_g}(\mu_1)-\frac{K}{2}t(1-t)W_2(\mu_0,\mu_1)^2 \end{align*}

for every $t\in[0,1]$.

In particular, $\operatorname{Ent}_{\operatorname{vol}_g}$ is $K$-displacement convex on $\mathcal P_2(M)$ in the following weak sense: for every pair of measures in $\mathcal P_2(M)$ with finite real entropy, there exists at least one constant-speed $W_2$-geodesic joining them along which the same entropy inequality holds.