Let $n\in\mathbb N$, let $(M,g)$ be a complete connected smooth $n$-dimensional Riemannian manifold without boundary, and let $V\in C^\infty(M;\mathbb R)$. Let $m$ be the Borel measure on $(M,\mathcal B(M))$ defined by
pointwise on $M$ for every $f\in C_c^\infty(M)$, and interpret $CD(K,\infty)$ as $K$-displacement convexity of the relative entropy along Wasserstein geodesics between finite-entropy endpoint measures. For finite $N\in(n,\infty)$, interpret $BE(K,N)$ as the condition that
pointwise on $M$ for every $f\in C_c^\infty(M)$, and interpret $CD(K,N)$ as the weak Lott-Sturm-Villani curvature-dimension condition for bounded-support absolutely continuous endpoint measures.
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Then the following assertions are equivalent:
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1. The metric measure space $(M,d_g,m)$ satisfies the corresponding smooth weighted Lott-Sturm-Villani condition $CD(K,N)$.
2. The operator $L$ satisfies the Bakry-Emery curvature-dimension condition $BE(K,N)$ on $C_c^\infty(M)$.
3. The tensor inequality
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\begin{align*}
\operatorname{Ric}_{V,N}\ge Kg
\end{align*}