On uniqueness of determination of a form of first degree by its integrals along geodesics

Abstract

We prove a theorem on uniqueness of determination of a form of first order inside some domain D by its integrals along geodesies of a fixed Riemannian metric connecting an arbitrary pair of points on the boundary of this domain. As an application, this result is used to study the problem of determining the coefficients at lower derivatives in a second order hyperbolic equation.

The X-Ray Transform for a Generic Family of Curves and Weights

“…In dimension n ≥ 3 there is no such known result for an arbitrary simple family of curves. On the other hand, if Γ is the family of the geodesics of a given (simple) Riemannian or Finsler metric, and w is close enough to a constant, injectivity and stability of I Γ,w was established in [Mu2,Mu3,AR,BG,R].…”

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

“…In this framework, the same questions (injectivity, stability, range characterization, reconstruction algorithms, inverse problems with partial data) as for the straight line case are still under active theoretical study. To the author's knowledge, numerical simulations for these transforms remain to be documented.When the metric is simple, injectivity over functions was proved in [12] and injectivity over solenoidal vector fields was established in [1,2]. Under the same simplicity assumption, the problem was recently proved in [17] to be injective over solenoidal tensors ("s-injective") of any order, and previously in [4] under assumptions on the curvature.…”

On the characterization of the kernel of the geodesic X-ray transform

“…We can use the formulas obtained above to compute the principal symbol of P P , in fact, we just need to replace G (1) , G (2) ,G (2) , G (3) in the equation above by g. Making the change of variables z = g −1/2 z , and denoting z again by z, we obtain…”

“…Anikonov and Romanov [1,2] have solved the case of 1-tensor fields. In the case of tensors of order 2, some partial answers have been given by Pestov and Sharafutdinov [10] in the case on negatively curved manifolds, which was extended to manifolds of small positive curvature by Sharafutdinov [11].…”