  We study the propagator of the wave equation on a closed Riemannian manifold
$M$. We propose a geometric approach to the construction of the propagator as a
single oscillatory integral global both in space and in time with a
distinguished complex-valued phase function. This enables us to provide a
global invariant definition of the full symbol of the propagator - a scalar
function on the cotangent bundle - and an algorithm for the explicit
calculation of its homogeneous components. The central part of the paper is
devoted to the detailed analysis of the subprincipal symbol; in particular, we
derive its explicit small time asymptotic expansion. We present a general
geometric construction that allows one to visualise topological obstructions
and describe their circumvention with the use of a complex-valued phase
function. We illustrate the general framework with explicit examples in
dimension two.
