In optical tomography one seeks to use near-infrared light to determine the optical absorption and scattering properties of a medium X ⊂ R^n. If the refractive index is constant throughout the medium, the steady-state case is modeled by the stationary linear transport equation in terms of the Euclidean metric. In this work we consider the case of variable refractive index where the dynamics are modeled by writing the transport equation in terms of a Riemannian metric; in the absence of interaction, photons follow the geodesics of this metric. In particular we study the problem where our measurements allow the application of an in-going flux depending on both position and direction, but we allow only a weighted average measurement of the out-going flux. We show that making measurements on all of bdry(X) determines the extinction coefficient and that once this is known, under additional assumptions, measurements at a single point on bdry(X) determine the scattering kernel.