The stationary linear transport
equation models the scattering and absorption of a low-density beam of neutrons as
it passes through a body. In Euclidean space, to a first approximation, particles
travel in straight lines. Here we study the analogous transport equation
for particles in an ambient field described by a Riemannian metric where,
again to first approximation, particles follow geodesics of the metric. We
consider the problem of determining the scattering and absorption coefficients
from knowledge of the albedo operator on the boundary of the domain.
Under certain restrictions, the albedo operator is shown to determine the
geodesic ray transform of the absorption coefficient; for “simple” manifolds this
transform is invertible and so the coefficient itself is determined. In dimensions 3
or greater, we show that one may then obtain the collision (or scattering)
kernel.
