Let $(M,g)$ be a complete connected finite-dimensional smooth Riemannian manifold without boundary. Let $d_g:M\times M\to[0,\infty)$ denote its Riemannian distance, and let $\operatorname{vol}_g$ denote its Riemannian volume measure on $(M,\mathcal B(M))$. Let $\mathcal P_2(M)$ be the set of Borel probability measures $\mu$ on $M$ such that, for one equivalently every $x_0\in M$, the integral $\int_M d_g(x,x_0)^2\,d\mu(x)$ is finite. For $\mu_0,\mu_1\in\mathcal P_2(M)$, define $W_2$ by

\begin{align*} W_2(\mu_0,\mu_1)^2:=\inf_{\pi\in\Pi(\mu_0,\mu_1)}\int_{M\times M} d_g(x,y)^2\,d\pi(x,y), \end{align*}

where $\Pi(\mu_0,\mu_1)$ is the set of Borel probability measures on $M\times M$ whose first marginal is $\mu_0$ and whose second marginal is $\mu_1$. Define $\operatorname{Ent}_{\operatorname{vol}_g}:\mathcal P_2(M)\to(-\infty,+\infty]$ as follows. If $\mu=\rho\operatorname{vol}_g$ and $\rho\log\rho\in L^1(M,\mathcal B(M),\operatorname{vol}_g)$, set

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

If $\mu$ is not absolutely continuous with respect to $\operatorname{vol}_g$, or if $\rho\log\rho\notin L^1(M,\mathcal B(M),\operatorname{vol}_g)$, set $\operatorname{Ent}_{\operatorname{vol}_g}(\mu)=+\infty$. For $K\in\mathbb R$, the following are equivalent:

1. The Ricci tensor satisfies $\operatorname{Ric}_p(v,v)\ge K g_p(v,v)$ for every $p\in M$ and every $v\in T_pM$. 2. For every $\mu_0,\mu_1\in\mathcal P_2(M)$ satisfying $\operatorname{Ent}_{\operatorname{vol}_g}(\mu_0)<\infty$ and $\operatorname{Ent}_{\operatorname{vol}_g}(\mu_1)<\infty$, there exists a constant-speed $W_2$-geodesic $(\mu_t)_{t\in[0,1]}$ from $\mu_0$ to $\mu_1$ such that, for every $t\in[0,1]$,

\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 K2 t(1-t)W_2(\mu_0,\mu_1)^2. \end{align*}