  This paper investigates some properties of entropy solutions of hyperbolic
conservation laws on a Riemannian manifold. First, we generalize the Total
Variation Diminishing (TVD) property to manifolds, by deriving conditions on
the flux of the conservation law and a given vector field ensuring that the
total variation of the solution along the integral curves of the vector field
is non-increasing in time. Our results are next specialized to the important
case of a flow on the 2-sphere, and examples of flux are discussed. Second, we
establish the convergence of the finite volume methods based on numerical
flux-functions satisfying monotonicity properties. Our proof requires detailed
estimates on the entropy dissipation, and extends to general manifolds an
earlier proof by Cockburn, Coquel, and LeFloch in the Euclidian case.
