Let $L=\DD+Z$ for a $C^1$ vector field $Z$ on a complete Riemannian manifoldpossibly with a boundary. By using the uniform distance, a number oftransportation-cost inequalities on the path space for the (reflecting)$L$-diffusion process are proved to be equivalent to the curvature condition$\Ric-\nn Z\ge - K$ and the convexity of the boundary (if exists). Theseinequalities are new even for manifolds without boundary, and are partlyextended to non-convex manifolds by using a conformal change of metric whichmakes the boundary from non-convex to convex.
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机译：让$ L = \ DD + Z $表示完整的黎曼流形上可能带有边界的$ C ^ 1 $向量场$ Z $。通过使用均匀距离，证明了（反射）$ L $-扩散过程在路径空间上的许多运输成本不等式等于曲率条件$ \ Ric- \ nn Z \ ge-K $和边界的凸度（如果存在）。这些不等式甚至对于无边界的流形都是新的，并且通过使用度量的共形变化将边界从非凸形变为凸形而部分扩展到非凸形流形。
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