Integral Geometry of Tensor Fields

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 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, (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

SOME QUESTIONS OF TENSOR ANALYSIS
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

Poincaré'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 A1

Proof 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