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01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information.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 of 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 of anisotropic media whose interaction with sounding radiation depends essentially on the direction in which the latter propagates. The main mathematical objects tackled have been given the term ''ray transform'' which refers mainly to optical and seismic rays rather than to X-rays.
Contents
INTRODUCTION The problem of determining a metric by its hodograph and a linearization of the problem The kinetic equation on a Riemannian manifold Some remarks 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, (x,y))Description of the image of the ray transform Integral moments of the function If 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 Application of the ray transform to an inverse problem of photoelasticity Further results Tensor fields Covariant differentiation Symmetric tensor fields Semibasic tensor fields The horizontal covariant derivative Formulas of Gauss--Ostrogradskii type for vertical and horizontal derivatives THE RAY TRANSFORM ON A RIEMANNIAN MANIFOLD Compact dissipative Riemannian manifolds The ray transform on a CDRM The problem of inverting the ray transform Pestov's differential identity Poincare's inequality for semibasic tensor fields Reduction of Theorem 4.3.3 to an inverse problem for the kinetic equation Proof of Theorem 4.3.3 Consequences for the nonlinear problem of determining a metric from its hodograph Bibliographical remarks THE TRANSVERSE RAY TRANSFORM Electromagnetic waves in quasi-isotropic media The transverse ray transform on a CDRM Reduction of Theorem 5.2.2 to an inverse problem for the kinetic equation Estimation of the summand related to the right-hand side of the kinetic equation Estimation of the boundary integral and summands depending on curvature Proof of Theorem 5.2.2 Decomposition of the operators A0 and A1Proof of Lemma 5.6.1 Final remarks THE TRUNCATED TRANSVERSE RAY TRANSFORM The polarization ellipse The truncated transverse ray transform Proof of Theorem 6.2.2 Decomposition of the operator Q, Proof of Lemma 6.3.1 Inversion of the truncated transverse ray transform on Euclidean space THE MIXED RAY TRANSFORM Elastic waves in quasi-isotropic media The mixed ray transform Proof of Theorem 7.2.2 The algebraic part of the proof THE EXPONENTIAL RAY TRANSFORM Formulation of the main definitions and results The modified horizontal derivative Proof of Theorem 8.1.1 The volume of a simple compact Riemannian manifold Determining a metric in a prescribed conformal class Bibliographical remarks Bibliography Index
