We describe all classes of inhomogeneous, transversely isotropic elastic media for which the sheets associated to each wave mode are ellipsoids. These media have the property that elastic waves in each mode propagate along geodesic segments of certain Riemannian metrics. We study the intersection of the sheets of the slowness surface for these media, and, in view of applications to the analysis of propagation of singularities along rays, we give pointwise conditions that guarantee that the sheet of the slowness surface corresponding to a given wave mode is disjoint from the others. We also investigate the smoothness of the associated polarization vectors as functions of position and direction. We employ coordinate and frame-independent methods, suitable to the study of the dynamic inverse boundary problem in elasticity.
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