  We consider the initial-value problem for stochastic continuity equations of
the form $$ \partial_t \rho + \text{div}_h \left[\rho \left(u(t,x) +
\sum_{i=1}^N a_i(x)\circ \frac{dW^i}{dt}\right)\right] = 0, $$ defined on a
smooth closed Riemanian manifold $M$ with metric $h$, where the Sobolev regular
velocity field $u$ is perturbed by Gaussian noise terms $\dot{W}_i(t)$ driven
by smooth spatially dependent vector fields $a_i(x)$ on $M$. Our main result is
that weak ($L^2$) solutions are renormalized solutions, that is, if $\rho$ is a
weak solution, then the nonlinear composition $S(\rho)$ is a weak solution as
well, for any "reasonable" function $S:\mathbb{R}\to\mathbb{R}$. The proof
consists of a systematic procedure for regularizing tensor fields on a
manifold, a convenient choice of atlas to simplify technical computations
linked to the Christoffel symbols, and several DiPerna-Lions type commutators
$\mathcal{C}_\varepsilon (\rho,D)$ between (first/second order) geometric
differential operators $D$ and the regularization device ($\varepsilon$ is the
scaling parameter). This work, which is related to the "Euclidean" result in
Punshon-Smith (2017), reveals some structural effects that noise and nonlinear
domains have on the dynamics of weak solutions.
