Let $n\in\mathbb N$, let $N\in(n,\infty)$, and let $(M,g)$ be a complete connected smooth $n$-dimensional Riemannian manifold without boundary. Let $\operatorname{vol}_g$ denote the Riemannian volume measure, let $V\in C^\infty(M;\mathbb R)$, and define the weighted Borel measure $m$ on $(M,\mathcal B(M))$ by

\begin{align*} d m=e^{-V}\,d\operatorname{vol}_g. \end{align*}

Assume that $m$ is locally finite and has full support. Define the $N$-Bakry-Emery Ricci tensor by

\begin{align*} \operatorname{Ric}_{V,N}:=\operatorname{Ric}_g+\operatorname{Hess}_g V-\frac{1}{N-n}\,dV\otimes dV. \end{align*}

If $K\in\mathbb R$ and

\begin{align*} \operatorname{Ric}_{V,N}(w,w)\ge K g_p(w,w) \end{align*}

for every $p\in M$ and every $w\in T_pM$, then the metric measure space $(M,d_g,m)$ satisfies the weak Lott-Sturm-Villani curvature-dimension condition $CD(K,N)$ for bounded-support absolutely continuous endpoint measures.

More explicitly, for every pair of Borel probability measures $\mu_0,\mu_1\in\mathcal P_2(M)$ with bounded support and $\mu_i=\rho_i m$ for Borel densities $\rho_i:M\to[0,\infty)$, there exist an optimal transport plan $\pi\in\Pi(\mu_0,\mu_1)$ for the quadratic cost $d_g^2$ and a $W_2$-geodesic $(\mu_t)_{t\in[0,1]}$ with $\mu_t=\rho_t m$ such that, for every $t\in[0,1]$,

\begin{align*} -\int_M \rho_t(x)^{1-\frac1N}\,d m(x)\le -\int_{M\times M}\left[\tau_{K,N}^{(1-t)}(d_g(x,y))\rho_0(x)^{-\frac1N}+\tau_{K,N}^{(t)}(d_g(x,y))\rho_1(y)^{-\frac1N}\right]\,d\pi(x,y), \end{align*}

with the standard Lott-Sturm-Villani distortion coefficients $\tau_{K,N}^{(t)}$.