Boundary and lens rigidity for non-convex manifolds

@article{Guillarmou2017BoundaryAL,
  title={Boundary and lens rigidity for non-convex manifolds},
  author={Colin Guillarmou and Marco Mazzucchelli and Leo Tzou},
  journal={arXiv: Differential Geometry},
  year={2017}
}
We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations… 
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References

SHOWING 1-10 OF 44 REFERENCES

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

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…

Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds

Let σ be the scattering relation on a compact Riemannian manifold M with nonnecessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the…

Boundary and lens rigidity of finite quotients

We consider compact Riemannian manifolds (M, ∂M, g) with boundary ∂M and metric g on which a finite group Γ acts freely. We determine the extent to which certain rigidity properties of (M, ∂M, g)…

Integral geometry problem for nontrapping manifolds

We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a…

Regularity of ghosts in tensor tomography

We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the…

The inverse problem for the local geodesic ray transform

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a…

Rigidity and the distance between boundary points

In this paper we consider some rigidity problems in Riemannian geometry. In particular, we prove Theorem A. Any complete Riemannian metric without conjugate points on R" which is isometric to the…

Spectral rigidity and invariant distributions on Anosov surfaces

This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on…

Recovery of the C∞ jet from the boundary distance function

For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the “boundary rigidity problem”. If the…

Lens rigidity for manifolds with hyperbolic trapped set

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $\gamma$ with endpoints on $\partial M$, together with their…