  We investigate a class of scalar conservation laws on manifolds driven by
multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian
manifold is shown to be well-posed. We prove existence of generalized kinetic
solutions using the vanishing viscosity method. A rigidity result is derived,
which implies that generalized solutions are kinetic solutions and that kinetic
solutions are uniquely determined by their initial data ($L^1$ contraction
principle). Deprived of noise, the equations we consider coincide with those
analyzed by Ben-Artzi and LeFloch (2007), who worked with Kruzkov-DiPerna
solutions. In the Euclidian case, the stochastic equations agree with those
examined by Debussche and Vovelle (2010).
