A geometric interpretation of the complex tensor Riccati equation for Gaussian beams

We study the complex Riccati tensor equation DċG+GCG−R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space. When time moves forward, Gaussian beams move along geodesics, and the Riccati equation determines the Hessian of the phase function for the Gaussian beam. The imaginary part of a solution G describes how a Gaussian beam decays in different directions of space. The main result of the present work is that the real part of G is the shape operator of the phase front for the Gaussian beam. This result generalizes a known result for the Riccati equation in R. The idea of the proof is to express the Riccati equation in Fermi coordinates adapted to the underlying geodesic. In Euclidean geometry we also study when the phase front is contained in the area of influence, or light cone.