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 de nition 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.
Consider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M , whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum of A decomposes, up to an error with superpolynomial decay, into m distinct series, each associated with one of the eigenvalues of the principal symbol of A. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose L 2 .M / into almost-orthogonal almost-invariant subspaces under the action of both A and the hyperbolic evolution.
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