Ricci curvature lower bounds on Sasakian manifolds

Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property $MCP(0,N)$ for some positive integer $N$. We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one.