  Let $L=\DD+Z$ for a $C^1$ vector field $Z$ on a complete Riemannian manifold
possibly with a boundary. By using the uniform distance, a number of
transportation-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). These
inequalities are new even for manifolds without boundary, and are partly
extended to non-convex manifolds by using a conformal change of metric which
makes the boundary from non-convex to convex.
