Boundary and lens rigidity for non-convex manifolds

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 surfa...

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Bibliographic Details
Published in:American journal of mathematics 2021-04, Vol.143 (2), p.533-575
Main Authors: Guillarmou, Colin, Mazzucchelli, Marco, Tzou, Leo
Format: Article
Language:English
Subjects:
Analysis of PDEs
Conjugate points
Conjugates
Convexity
Differential Geometry
Domains
Lenses
Mathematics
Riemann surfaces
Rigidity
Tensors
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Summary: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.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2021.0012