Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds

Abstract

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 $\mathrm{CD}(K,\infty )$ condition and non-collapsed $\mathrm{CD}(K,N)$ condition on Riemannian manifolds with boundary. In particular, using $L^1$-optimal transportation theory, we prove that $\mathrm{CD}(K,\infty )$ condition implies geodesical convexity.

References in corpus (2)

• On the structure of RCD spaces with upper curvature bounds
• Ricci tensor on smooth metric measure space with boundary

Cited by in corpus (4)

• New differential operator and non-collapsed $RCD$ spaces
• Weakly non-collapsed RCD spaces are strongly non-collapsed
• On the structure of RCD spaces with upper curvature bounds
• Rigidity of some functional inequalities on RCD spaces