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In this paper we investigate Lott-Sturm-Villani's synthetic lower Riccicurvature bound on Riemannian manifolds with boundary. We prove several measurerigidity 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. Inparticular, using $L^1$-optimal transportation theory, we prove that ${\rmCD}(K, \infty)$ condition implies geodesical convexity.

Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds