  In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci
curvature bound on Riemannian manifolds with boundary. We prove several measure
rigidity results for some important functional and geometric inequalities,
which completely characterize ${\rm CD}(K, \infty)$ condition and non-collapsed
${\rm CD}(K, N)$ condition on Riemannian manifolds with boundary. In
particular, using $L^1$-optimal transportation theory, we prove that ${\rm
CD}(K, \infty)$ condition implies geodesical convexity.
