  The aim of this note is to provide regularity results for Regular Lagrangian
flows of Sobolev vector fields over compact metric measure spaces verifying the
Riemannian curvature dimension condition. We first prove, borrowing some ideas
already present in the literature, that flows generated by vector fields with
bounded symmetric derivative are Lipschitz, providing the natural extension of
the standard Cauchy-Lipschitz theorem to this setting. Then we prove a
Lusin-type regularity result in the Sobolev case (under the additional
assumption that the m.m.s. is Ahlfors regular) therefore extending the already
known Euclidean result.
