Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge

Abstract

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M\times \partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suitable neighborhood of $p$ in $M$, up to the natural diffeomorphism invariance of the problem.

We also consider the closely related lens rigidity problem which is a more natural formulation if the boundary distance is not realized by unique minimizing geodesics. The lens relation measures the point and the direction of exit from $M$ of geodesics issued from the boundary and the length of the geodesic. The lens rigidity problem is whether we can determine the metric up to isometry from the lens relation. We solve the lens rigidity problem under the assumption that there is a function on $M$ with suitable convexity properties relative to $g$. This can be considered as a complete solution of a problem formulated first by Herglotz in 1905. We also prove a semi-global results given semi-global data. This shows, for instance, that simply connected manifolds with strictly convex boundaries are lens rigid if the sectional curvature is non-positive or non-negative or if there are no focal points.

The key tool is the analysis of the geodesic X-ray transform on 2-tensors, corresponding to a metric $g$, in the normal gauge, such as normal coordinates relative to a hypersurface, where one also needs to allow weights. This is handled by refining and extending our earlier results in the solenoidal gauge.

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      mrclass = {53C24 (47G30 53C21 53C65)},
      mrnumber = {2414415},
      mrreviewer = {Rodney Josué Biezuner},
      url = {http://www.math.bas.bg/serdica/2008/2008-067-112.pdf},
      zblnumber = {1199.53099},
      }
  • [SU-Duke] Go to document P. Stefanov and G. Uhlmann, "Stability estimates for the X-ray transform of tensor fields and boundary rigidity," Duke Math. J., vol. 123, iss. 3, pp. 445-467, 2004.
    @ARTICLE{SU-Duke,
      author = {Stefanov, Plamen and Uhlmann, Gunther},
      title = {Stability estimates for the {X}-ray transform of tensor fields and boundary rigidity},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {123},
      year = {2004},
      number = {3},
      pages = {445--467},
      issn = {0012-7094},
      mrclass = {53C65 (44A12 47G30 53C24)},
      mrnumber = {2068966},
      mrreviewer = {Dorothee Schueth},
      doi = {10.1215/S0012-7094-04-12332-2},
      url = {https://doi.org/10.1215/S0012-7094-04-12332-2},
      zblnumber = {1058.44003},
      }
  • [SU-JAMS] Go to document P. Stefanov and G. Uhlmann, "Boundary rigidity and stability for generic simple metrics," J. Amer. Math. Soc., vol. 18, iss. 4, pp. 975-1003, 2005.
    @ARTICLE{SU-JAMS,
      author = {Stefanov, Plamen and Uhlmann, Gunther},
      title = {Boundary rigidity and stability for generic simple metrics},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {18},
      year = {2005},
      number = {4},
      pages = {975--1003},
      issn = {0894-0347},
      mrclass = {53C24 (58J32)},
      mrnumber = {2163868},
      mrreviewer = {Rodney Josué Biezuner},
      doi = {10.1090/S0894-0347-05-00494-7},
      url = {https://doi.org/10.1090/S0894-0347-05-00494-7},
      zblnumber = {1079.53061},
      }
  • [SU-Kawai] Go to document P. Stefanov and G. Uhlmann, "Boundary and lens rigidity, tensor tomography and analytic microlocal analysis," in Algebraic Analysis of Differential Equations from Microlocal Analysis to Exponential Asymptotics, Springer, Tokyo, 2008, pp. 275-293.
    @INCOLLECTION{SU-Kawai,
      author = {Stefanov, Plamen and Uhlmann, Gunther},
      title = {Boundary and lens rigidity, tensor tomography and analytic microlocal analysis},
      booktitle = {Algebraic Analysis of Differential Equations from Microlocal Analysis to Exponential Asymptotics},
      pages = {275--293},
      publisher = {Springer, Tokyo},
      year = {2008},
      mrclass = {58J40 (47G30)},
      mrnumber = {2758914},
      mrreviewer = {Alberto Parmeggiani},
      doi = {10.1007/978-4-431-73240-2\_23},
      url = {https://doi.org/10.1007/978-4-431-73240-2_23},
      zblnumber = {1138.53039},
      }
  • [SU-MRL] Go to document P. Stefanov and G. Uhlmann, "Rigidity for metrics with the same lengths of geodesics," Math. Res. Lett., vol. 5, iss. 1-2, pp. 83-96, 1998.
    @ARTICLE{SU-MRL,
      author = {Stefanov, Plamen and Uhlmann, Gunther},
      title = {Rigidity for metrics with the same lengths of geodesics},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {5},
      year = {1998},
      number = {1-2},
      pages = {83--96},
      issn = {1073-2780},
      mrclass = {53C22 (53C20)},
      mrnumber = {1618347},
      mrreviewer = {Raul Quiroga-Barranco},
      doi = {10.4310/MRL.1998.v5.n1.a7},
      url = {https://doi.org/10.4310/MRL.1998.v5.n1.a7},
      zblnumber = {0934.53031},
      }
  • [SU-lens] Go to document P. Stefanov and G. Uhlmann, "Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds," J. Differential Geom., vol. 82, iss. 2, pp. 383-409, 2009.
    @ARTICLE{SU-lens,
      author = {Stefanov, Plamen and Uhlmann, Gunther},
      title = {Local lens rigidity with incomplete data for a class of non-simple {R}iemannian manifolds},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {82},
      year = {2009},
      number = {2},
      pages = {383--409},
      issn = {0022-040X},
      mrclass = {53C24},
      mrnumber = {2520797},
      doi = {10.4310/jdg/1246888489},
      url = {https://doi.org/10.4310/jdg/1246888489},
      zblnumber = {1247.53049},
      }
  • [SUV_localrigidity] Go to document P. Stefanov, G. Uhlmann, and A. Vasy, "Boundary rigidity with partial data," J. Amer. Math. Soc., vol. 29, iss. 2, pp. 299-332, 2016.
    @ARTICLE{SUV_localrigidity,
      author = {Stefanov, Plamen and Uhlmann, Gunther and Vasy, Andras},
      title = {Boundary rigidity with partial data},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {29},
      year = {2016},
      number = {2},
      pages = {299--332},
      issn = {0894-0347},
      mrclass = {53C65 (35R30 53C24)},
      mrnumber = {3454376},
      mrreviewer = {V. K. Ohanyan},
      doi = {10.1090/jams/846},
      url = {https://doi.org/10.1090/jams/846},
      zblnumber = {1335.53055},
      }
  • [SUV_elastic] Go to document P. Stefanov, G. Uhlmann, and A. Vasy, "Local recovery of the compressional and shear speeds from the hyperbolic DN map," Inverse Problems, vol. 34, iss. 1, p. 014003, 2018.
    @ARTICLE{SUV_elastic,
      author = {Stefanov, Plamen and Uhlmann, Gunther and Vasy, Andras},
      title = {Local recovery of the compressional and shear speeds from the hyperbolic {DN} map},
      journal = {Inverse Problems},
      fjournal = {Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data},
      volume = {34},
      year = {2018},
      number = {1},
      pages = {014003, 13},
      issn = {0266-5611},
      mrclass = {35Q74 (74J25)},
      mrnumber = {3742360},
      mrreviewer = {Giuseppe Saccomandi},
      doi = {10.1088/1361-6420/aa9833},
      url = {https://doi.org/10.1088/1361-6420/aa9833},
      zblnumber = {06850137},
      }
  • [SUV:Tensor] Go to document P. Stefanov, G. Uhlmann, and A. Vasy, "Inverting the local geodesic X-ray transform on tensors," J. Anal. Math., vol. 136, iss. 1, pp. 151-208, 2018.
    @ARTICLE{SUV:Tensor,
      author = {Stefanov, Plamen and Uhlmann, Gunther and Vasy, Andr\'{a}s},
      title = {Inverting the local geodesic {X}-ray transform on tensors},
      journal = {J. Anal. Math.},
      fjournal = {Journal d'Analyse Mathématique},
      volume = {136},
      year = {2018},
      number = {1},
      pages = {151--208},
      issn = {0021-7670},
      mrclass = {53C65},
      mrnumber = {3892472},
      mrreviewer = {B. S. Rubin},
      doi = {10.1007/s11854-018-0058-3},
      url = {https://doi.org/10.1007/s11854-018-0058-3},
      zblnumber = {07008552},
      }
  • [TriggianiY] Go to document R. Triggiani and P. F. Yao, "Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot," Appl. Math. Optim., vol. 46, iss. 2-3, pp. 331-375, 2002.
    @ARTICLE{TriggianiY,
      author = {Triggiani, Roberto and Yao, P. F.},
      title = {Carleman estimates with no lower-order terms for general {R}iemann wave equations. {G}lobal uniqueness and observability in one shot},
      note = {special issue dedicated to the memory of Jacques-Louis Lions},
      journal = {Appl. Math. Optim.},
      fjournal = {Applied Mathematics and Optimization},
      volume = {46},
      year = {2002},
      number = {2-3},
      pages = {331--375},
      issn = {0095-4616},
      mrclass = {93C20 (35B45 35L70 58J45 93B07)},
      mrnumber = {1944764},
      mrreviewer = {Paolo Albano},
      doi = {10.1007/s00245-002-0751-5},
      url = {https://doi.org/10.1007/s00245-002-0751-5},
      zblnumber = {1030.35018},
      }
  • [UV:local] Go to document G. Uhlmann and A. Vasy, "The inverse problem for the local geodesic ray transform," Invent. Math., vol. 205, iss. 1, pp. 83-120, 2016.
    @ARTICLE{UV:local,
      author = {Uhlmann, Gunther and Vasy, Andr\'{a}s},
      title = {The inverse problem for the local geodesic ray transform},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {205},
      year = {2016},
      number = {1},
      pages = {83--120},
      issn = {0020-9910},
      mrclass = {53C65 (35R30 35S05 53C21)},
      mrnumber = {3514959},
      mrreviewer = {Gaoyong Zhang},
      doi = {10.1007/s00222-015-0631-7},
      url = {https://doi.org/10.1007/s00222-015-0631-7},
      zblnumber = {1350.53098},
      }
  • [V] Go to document J. Vargo, "A proof of lens rigidity in the category of analytic metrics," Math. Res. Lett., vol. 16, iss. 6, pp. 1057-1069, 2009.
    @ARTICLE{V,
      author = {Vargo, James},
      title = {A proof of lens rigidity in the category of analytic metrics},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {16},
      year = {2009},
      number = {6},
      pages = {1057--1069},
      issn = {1073-2780},
      mrclass = {53C24 (53C22)},
      mrnumber = {2576693},
      mrreviewer = {Rodney Josué Biezuner},
      doi = {10.4310/MRL.2009.v16.n6.a13},
      url = {https://doi.org/10.4310/MRL.2009.v16.n6.a13},
      zblnumber = {1202.53041},
      }
  • [Vasy:Minicourse] Go to document A. Vasy, "A minicourse on microlocal analysis for wave propagation," in Asymptotic Analysis in General Relativity, Cambridge Univ. Press, Cambridge, 2018, vol. 443, pp. 219-374.
    @INCOLLECTION{Vasy:Minicourse,
      author = {Vasy, Andr\'{a}s},
      title = {A minicourse on microlocal analysis for wave propagation},
      booktitle = {Asymptotic {A}nalysis in {G}eneral {R}elativity},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {443},
      pages = {219--374},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2018},
      mrclass = {58J45 (58J40 58J50)},
      mrnumber = {3792086},
      mrreviewer = {Matteo Capoferri},
      doi = {10.1017/9781108186612.005},
      url = {https://doi.org/10.1017/9781108186612.005},
      zblnumber = {1416.83026},
      }
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    @ARTICLE{Vasy-Dyatlov:Microlocal-Kerr,
      author = {Vasy, Andr\'{a}s},
      title = {Microlocal analysis of asymptotically hyperbolic and {K}err-de {S}itter spaces (with an appendix by {S}emyon {D}yatlov)},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {194},
      year = {2013},
      number = {2},
      pages = {381--513},
      issn = {0020-9910},
      mrclass = {58J47 (35R01 83C57)},
      mrnumber = {3117526},
      mrreviewer = {Davide Batic},
      doi = {10.1007/s00222-012-0446-8},
      url = {https://doi.org/10.1007/s00222-012-0446-8},
      zblnumber = {1315.35015},
      }
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    @ARTICLE{Wen_2015,
      author = {Wen, Haomin},
      title = {Simple {R}iemannian surfaces are scattering rigid},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {19},
      year = {2015},
      number = {4},
      pages = {2329--2357},
      issn = {1465-3060},
      mrclass = {53C24 (53C22 57M50)},
      mrnumber = {3375529},
      mrreviewer = {Inkang Kim},
      doi = {10.2140/gt.2015.19.2329},
      url = {https://doi.org/10.2140/gt.2015.19.2329},
      zblnumber = {1323.53041},
      }
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    @ARTICLE{WZ,
      author = {Wiechert, E. and Zoeppritz, K.},
      title = {{Ü}ber {E}rdbebenwellen},
      journal = {Nachr. Koenigl. Geselschaft Wiss. Göttingen},
      volume = {4},
      pages = {415--549},
      year = {1907},
      jfmnumber = {38.0970.01},
      }

Authors

Plamen Stefanov

Purdue University, West Lafayette, IN

Gunther Uhlmann

University of Washington, Seattle, WA and Institute for Advanced Study, HKUST, Clear Water Bay, Hong Kong, China

András Vasy

Stanford University, Stanford, CA