  We consider nonlinear hyperbolic conservation laws, posed on a differential
(n+1)-manifold with boundary referred to as a spacetime, and in which the
"flux" is defined as a flux field of n-forms depending on a parameter (the
unknown variable). We introduce a formulation of the initial and boundary value
problem which is geometric in nature and is more natural than the vector field
approach recently developed for Riemannian manifolds. Our main assumption on
the manifold and the flux field is a global hyperbolicity condition, which
provides a global time-orientation as is standard in Lorentzian geometry and
general relativity. Assuming that the manifold admits a foliation by compact
slices, we establish the existence of a semi-group of entropy solutions.
Moreover, given any two hypersurfaces with one lying in the future of the
other, we establish a "contraction" property which compares two entropy
solutions, in a (geometrically natural) distance equivalent to the L1 distance.
To carry out the proofs, we rely on a new version of the finite volume method,
which only requires the knowledge of the given n-volume form structure on the
(n+1)-manifold and involves the {\sl total flux} across faces of the elements
of the triangulations, only, rather than the product of a numerical flux times
the measure of that face.
