# American Institute of Mathematical Sciences

December  2018, 12(6): 1365-1387. doi: 10.3934/ipi.2018057

## Lens rigidity with partial data in the presence of a magnetic field

 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK

* Current address: Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106-3080, USA

Received  November 2017 Revised  June 2018 Published  October 2018

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension $≥ 3$ with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magnetic geodesics) boundary point where the lens data is accessible. We also prove a boundary rigidity result with partial data assuming the lengths of magnetic geodesics joining boundary points near a strictly convex boundary point are known. The local lens rigidity result also leads to a global rigidity result under some strictly convex foliation condition. A discussion of a weaker version of the lens rigidity problem with partial data for general smooth curves is given at the end of the paper.

Citation: Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems & Imaging, 2018, 12 (6) : 1365-1387. doi: 10.3934/ipi.2018057
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