Integral Geometry of Tensor Fields on a Class of Non-Simple Riemannian Manifolds

By Plamen Stefanov, Gunther Uhlmann. 2006

87 citations recorded by the source corpus.

We study the geodesic X-ray transform IΓ of tensor fields on a compact Riemannian manifold M with non-necessarily convex boundary and with possible conjugate points. We assume that IΓ is known for geodesics belonging to an open set Γ with endpoints on the boundary. We prove generic s-injectivity and a stability estimate under some topological assumptions and under the condition that for any (x, ξ) ∈ T*M, there is a geodesic in Γ through x normal to ξ without conjugate points.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/951802f86f8f2f6bd4e7366bb2cf29d477bab3d5

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In Astronomy and Astrophysics

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

The geodesic X-ray transform with fold caustics (2010, background): “M /; for all k 0, see [23, 9], and [25] for a class of manifolds with conjugate points.”
The X-Ray Transform for a Generic Family of Curves and Weights (2007, result): “The arguments that follow are close to those in [ 22 ].”
Inverse problems: seeing the unseen (2014, background): “The general case is considered in [182], and is based on an application of a complex stationary phase argument [173] to (110) as in [107].”
A Support Theorem for the Geodesic Ray Transform on Functions (2007, background): “One needs additional restrictions on the metric even to prove injectivity results for this transform as the following counterexample shows [16]: Consider the unit sphere with a small disk excised out from its east pole making the resulting manifold a smooth manifold with boundary.”
The Calderón problem in transversally anisotropic geometries (2013, background): “(c) A class of non-simple manifolds of any dimension such that there are sufficiently many geodesics without conjugate points and the metric is close to a real-analytic one (see [39] for the precise description of this class).”
Sensitivity analysis of an inverse problem for the wave equation with caustics (2012, methodology): “We refer to [28], [30] and the references therein for useful boundary estimates.”
Mathematics of thermoacoustic tomography (2007, background): “This, for instance, can be extracted from the results of [117].”