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

Abstract

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.

A Support Theorem for the Geodesic Ray Transform on Functions

“…See the book of J. Sjöstrand [14] for more details. Actually Stefanov-Uhlmann in [16,Prop.2] have proved the statement of the proposition with f replaced by a solenoidal symmetric tensor field. Their proof carries over here with straightforward modifications and in fact the arguments for functions are simpler.…”

“…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. Now consider a function f = 1 on a small disk centered at the north pole and f = −1 on a symmetrical disk centered at the south pole.…”

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

“…The problem for generic metrics is solved in [19,20]. In [21], the linear problem is considered under some assumption that is weaker than the simplicity. There is also a couple of results for manifolds with nonconvex boundary [16,6].…”

Inverse problems: seeing the unseen

“…We refer to [182] for more details about the results in this section. It turns out that a linearization of the lens rigidity problem is again the problem of s-injectivity of the ray transform I .…”

“…We note that we will also consider the case of incomplete data, that is when we don't have information about all the geodesics entering the manifold. More details can be found in [182,183].…”