  We prove a regularity result for Lagrangian flows of Sobolev vector fields
over RCD(K,N) metric measure spaces, regularity is understood with respect to a
newly defined quasi-metric built from the Green function of the Laplacian. Its
main application is that RCD(K,N) spaces have constant dimension. In this way
we generalize to such abstract framework a result proved by Colding-Naber for
Ricci limit spaces, introducing ingredients that are new even in the smooth
setting.
