  On gas giant planets the speed of sound is isotropic and goes to zero at the
surface. Geometrically, this corresponds to a Riemannian manifold whose metric
tensor has a conformal blow-up near the boundary. The blow-up is tamer than in
asymptotically hyperbolic geometry: the boundary is at a finite distance.
  We study the differential geometry of such manifolds, especially the
asymptotic behavior of geodesics near the boundary. We relate the geometry to
the propagation of singularities of a hydrodynamic PDE and we give the basic
properties of the Laplace--Beltrami operator. We solve two inverse problems,
showing that the interior structure of a gas giant is uniquely determined by
different types of boundary data.
