  We analyse the massless wave equation on a class of two dimensional manifolds
consisting of an arbitrary number of topological cylinders connected to one or
more topological spheres. Such manifolds are endowed with a degenerate
(non-globally hyperbolic) metric. Attention is drawn to the topological
constraints on solutions describing monochromatic modes on both compact and
non-compact manifolds. Energy and momentum currents are constructed and a new
global sum rule discussed. The results offer a rigorous background for the
formulation of a field theory of topologically induced particle production.
