Geometric wave propagator on Riemannian manifolds (Q6172847)

Wave equations for the Laplacian on closed Riemannian manifolds can be solved, in principle, imitating the formulas valid in the classical Euclidean case. However when looking for an explicit expression of the propagator by oscillatory integrals, one has to tackle with difficult problems due to the presence of caustics. Here the authors follow the approach of \textit{A. Laptev} et al. [Commun. Pure Appl. Math. 47, No. 11, 1411--1456 (1994; Zbl 0811.35177)] writing the propagator in terms of a global single Fourier integral operator, using complex-valued phase-functions. The authors go further giving more explicit results by geometric arguments. Namely, the invariant expressions of the global symbols and sub-principal symbols are provided using the notion of Levi-Civita phase function as a basic ingredient. A relevant application is the calculation of higher-order Weyl coefficients. In the last part of the paper, the authors apply their construction to explicit examples in dimension 2: the sphere and the hyperbolic plane.