Communications in Analysis and Geometry

Volume 30 (2022)

Number 8

Geometric wave propagator on Riemannian manifolds

Pages: 1713 – 1777



Matteo Capoferri (Department of Mathematics, University College London, United Kingdom; and Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom)

Michael Levitin (Department of Mathematics and Statistics, University of Reading, United Kingdom)

Dmitri Vassiliev (Department of Mathematics, University College London, United Kingdom)


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.

Received 15 May 2019

Accepted 11 March 2020

Published 13 July 2023