  In this paper we show existence of a trace for functions of bounded variation
on Riemannian manifolds with boundary. The trace, which is bounded in
$L^\infty$, is reached via $L^1$-convergence and allows an integration by parts
formula. We apply these results in order to show well-posedness and total
variation estimates for the initial boundary value problem for a scalar
conservation law on compact Riemannian manifolds with boundary in the context
of functions of bounded variation via the vanishing viscosity method. The flux
function is assumed to be time-dependent and divergence-free.
