We revise a uniqueness question for the scalar conservation law with stochastic forcing du + divɡ f(x, u)dt = Φ(x, u)dW t , x ∈ M, t ≥ 0 on a smooth compact Riemannian manifold (M,) where W t is the Wiener process and x → f(x, ξ) is a vector field on M for each ξ ∈ R. We introduce admissibility conditions, derive the kinetic formulation and use it to prove uniqueness in a more straightforward way than in the existing literature.