Integral geometry can be defined as determining some function or a more general quantity, which is defined on a manifold, given its integrals over submanifolds or a prescribed class. In this book, only integral geometry problems are considered for which the submanifolds are one-dimensional. The book deals with integral geometry of symmetric tensor fields. This section of integral geometry can be considered as the mathematical basis for tomography or anisotropic media whose interaction with sounding radiation depends essentially on the direction in which the latter propagates. The main mathematical objects tackled have been give the term "ray transform", which refers mainly to optical and seismic rays rather than to X-rays. This book should be of interest to mathematicians, engineers, computer scientists and physicists who are woking in the fields of light-conductor technology, plasma physics tomography and liquid crystals.
Introduction: the problem of determining a metric by its hodograph and a linearization of the problem; the kinetic equation in a Riemannian manifold. Part 1 The ray transform of symmetric tensor fields on Euclidean space: the ray transform and its relationship to the Fourier transform; description of the kernel of the ray transform in the smooth case; equivalence of the first two statements of theorem 2.2.1 in the case n=2; proof of theorem 2.2.2.; the ray transform of a field-distribution; decomposition of a tensor field into potential and solenoidal parts; a theorem on the tangent component; a theorem on conjugate tensor fields on the sphere; primality of the ideal ([x]2, ); description of the image of the ray transform; integral moments of the function I f; inversion formulas for the ray transform; proof of theorem 2.12.1; inversion of the ray transform on the space of field-distributions; the Plancherel formula for the ray transform; applications of the ray transform to an inverse problem of photoelasticity; further results. Part 2 Some questions of tensor analysis. Part 3 The ray transform on a Riemannian manifold. Part 4 The transverse ray transform. Part 5 The truncated transverse ray transform. Part 6 The mixed ray transform. Part 7 The exponential ray transform (Part contents)