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Ray Transform on Riemannian Manifolds

In: New Analytic and Geometric Methods in Inverse Problems

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  • Vladimir A. Sharafutdinov

    (Sobolev Institute of Mathematics)

Abstract

What is integral geometry? Since the famous paper by I. Radon in 1917, it has been agreed that integral geometry problems consist in determining sonic function or a more general object (cohomology class, tensor field, etc.) on a manifold, given its integrals over submanifolds of a prescribed class. In these lectures we only consider integral geometry problems for which the above-mentioned submanifolds are one-dimensional. Strictly speaking, the latter are always geodesics of a fixed Riemannian metric, in particular straight lines in Euclidean space. The exception is Lecture 1 in which we consider an arbitrary regular family of curves in a two-dimensional domain.

Suggested Citation

  • Vladimir A. Sharafutdinov, 2004. "Ray Transform on Riemannian Manifolds," Springer Books, in: Kenrick Bingham & Yaroslav V. Kurylev & Erkki Somersalo (ed.), New Analytic and Geometric Methods in Inverse Problems, pages 187-259, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-08966-8_6
    DOI: 10.1007/978-3-662-08966-8_6
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