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

Plamen Stefanov Purdue University Gunther Uhlmann University of Washington

Differential Geometry mathscidoc:1609.10121

Journal of Differential Geometry, 82, (2), 383-409, 2009
Let be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let . be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of 冃 and . restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T M; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that 冃 and . restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.
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@inproceedings{plamen2009local,
  title={LOCAL LENS RIGIDITY WITH INCOMPLETE DATA FOR A CLASS OF NON-SIMPLE RIEMANNIAN MANIFOLDS},
  author={Plamen Stefanov, and Gunther Uhlmann},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911184019964911778},
  booktitle={Journal of Differential Geometry},
  volume={82},
  number={2},
  pages={383-409},
  year={2009},
}
Plamen Stefanov, and Gunther Uhlmann. LOCAL LENS RIGIDITY WITH INCOMPLETE DATA FOR A CLASS OF NON-SIMPLE RIEMANNIAN MANIFOLDS. 2009. Vol. 82. In Journal of Differential Geometry. pp.383-409. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911184019964911778.
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