  Gaussian beams describe the amplitude and phase of rays and are widely used
to model acoustic propagation. This paper describes four new results in the
theory of Gaussian beams. (1) A new version of the \v{C}erven\'y equations for
the amplitude and phase of Gaussian beams is developed by applying the
equivalence of Hamilton-Jacobi theory with Jacobi's equation that connects
Riemannian curvature to geodesic flow. Thus the paper makes a fundamental
connection between Gaussian beams and an acoustic channel's so-called intrinsic
Gaussian curvature from differential geometry. (2) A new formula
$\pi(c/c")^{1/2}$ for the distance between convergence zones is derived and
applied to several well-known profiles. (3) A class of "model spaces" are
introduced that connect the acoustics of ducting/divergence zones with the
channel's Gaussian curvature $K=cc"-(c')^2$. The "model" SSPs yield constant
Gaussian curvature in which the geometry of ducts corresponds to great circles
on a sphere and convergence zones correspond to antipodes. The distance between
caustics $\pi(c/c")^{1/2}$ is equated with an ideal hyperbolic cosine SSP duct.
(4) An "intrinsic" version of \v{C}erven\'y's formulae for the amplitude and
phase of Gaussian beams is derived that does not depend on an "extrinsic"
arbitrary choice of coordinates such as range and depth. Direct comparisons are
made between the computational frameworks used by the three different
approaches to Gaussian beams: Snell's law, the extrinsic Frenet-Serret
formulae, and the intrinsic Jacobi methods presented here. The relationship of
Gaussian beams to Riemannian curvature is explained with an overview of the
modern covariant geometric methods that provide a general framework for
application to other special cases.
