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In this article we prove that stochastic differential equation (SDE) withSobolev drift on compact Riemannian manifold admits a unique $\nu$-almosteverywhere stochastic invertible flow, where $\nu$ is the Riemannian measure,which is quasi-invariant with respect to $\nu$. In particular, we extend thewell known DiPerna-Lions flows of ODEs to SDEs on Riemannian manifold.

Quasi Invariant Stochastic Flows of SDEs with Non-smooth Drifts on Riemannian Manifolds$^*$