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)

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Integral Geometry of Tensor Fields

Let (M, g) be a Riemannian manifold with boundary. Denote by pg the distance function in the metric g. We consider the inverse problem of whether pg(x,y), known for all x, y on <9M, determines the metric uniquely. This problem arose in geophysics in an attempt to determine the inner structure of the Earth by measuring the travel times of seismic waves. It goes back to Herglotz [H] and Wiechert and Zoeppritz [WZ]. Although the emphasis has been in the case that the medium is isotropic, the anisotropic case has been of interest in geophysics since it has been found that the inner core of the Earth exhibits anisotropic behavior [Cr]. In differential geometry this inverse problem has been studied because of rigidity questions and is known as the boundary rigidity problem. It is clear that one cannot determine the metric uniquely. Any isometry which is the identity at the boundary will give rise to the same measurements. Furthermore, the boundary distance function only takes into account the shortest paths, and it is easy to find counterexamples to unique determination, so one needs to pose some restrictions on the metric. Michel [Mi] conjectured that a simple metric g is uniquely determined, up to an action of a diffeomorphism fixing the boundary, by the boundary distance function pg(x,y) known for all x and y on dM. We recall Definition 1.1. We say that the Riemannian metric g is simple in M, if dM is strictly convex w.r.t. g, and for any x G M, the exponential map exp^ : exp~1(M) ? M is a diffeomorphism.

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Boundary rigidity and stability for generic simple metrics

We study the boundary rigidity problem for domains in Rn: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρg(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform Ig. We prove that the normal operator Ng=I*gIg is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

Stability estimates for the X-ray transform of tensor fields and boundary rigidity

This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderon’s problem since the famous analyst proposed it in the mathematical literature (Calderon in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. 65–73, 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?

V. A. Sharafutdinov

Papers

Integral geometry of tensor fields on a manifold of negative curvature