Geometric wave propagator on Riemannian manifolds

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 obstructions due to caustics and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.