The X-ray Transform on a General Family of Curves on Finsler Surfaces

Abstract

For a compact oriented Finsler surface with smooth boundary, we consider the scalar and vector integral geometry problems over a general family of curves running between boundary points and parametrized by arc length. We impose a natural condition which results in the no conjugate points condition in the case when the curves in question are geodesic lines. Our main theorem generalizes Mukhometov’s theorem in several directions. We also consider these problems on a closed oriented Finsler surface. In this case the integral geometry problems make sense provided that sufficiently many curves in the family are periodic. To this end, we assume that the induced flow on the unit circle bundle of the surface is Anosov. Also, we study the cohomological equation of thermostats without conjugate points.