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Boundary and lens rigidity for non-convex manifolds
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 143, Number 2, April 2021
- pp. 533-575
- 10.1353/ajm.2021.0012
- Article
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abstract:
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, allowing a non-empty trapped set.