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
References:
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C. Croke, Boundary and lens rigidity of finite quotients, Proc. Am. Math. Soc., 133 (2005), 3663-3668.  doi: 10.1090/S0002-9939-05-07927-X.  Google Scholar

[9]

C. Croke, Scattering rigidity with trapped geodesics, Ergodic Theory Dynam. Systems, 34 (2014), 826-836.  doi: 10.1017/etds.2012.164.  Google Scholar

[10]

C. Croke and B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector field, J. Diff. Geom., 39 (1994), 659-680.  doi: 10.4310/jdg/1214455076.  Google Scholar

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N. S. Dairbekov, Integral geometry problem for nontrapping manifolds, Inverse Problems, 22 (2006), 431-445.  doi: 10.1088/0266-5611/22/2/003.  Google Scholar

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N. S. DairbekovG. P. PaternainP. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609.  doi: 10.1016/j.aim.2007.05.014.  Google Scholar

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N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Problems and Imaging, 4 (2010), 397-409.  doi: 10.3934/ipi.2010.4.397.  Google Scholar

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B. FrigyikP. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108.  doi: 10.1007/s12220-007-9007-6.  Google Scholar

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C. Guillarmou, Lens rigidity for manifolds with hyperbolic trapped set, J. Amer. Math. Soc., 30 (2017), 561-599.  doi: 10.1090/jams/865.  Google Scholar

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G. Herglotz, Über die Elastizität der Erde bei Berücksichtigung ihrer variablen Dichte, Zeitschr. für Math. Phys., 52 (1905), 275-299.   Google Scholar

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P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528.  doi: 10.4310/CAG.2012.v20.n3.a3.  Google Scholar

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P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664.  doi: 10.1007/s12220-010-9162-z.  Google Scholar

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S. Holman and P. Stefanov, The weighted Doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130.  doi: 10.3934/ipi.2010.4.111.  Google Scholar

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V. Krishnan, A support theorem for the geodesic ray transform on functions, J. Fourier Anal. Appl., 15 (2009), 515-520.  doi: 10.1007/s00041-009-9061-5.  Google Scholar

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V. Krishnan and P. Stefanov, A support theorem for the geodesic ray transform of symmetric tensor fields, Inverse Problems and Imaging, 3 (2009), 453-464.  doi: 10.3934/ipi.2009.3.453.  Google Scholar

[25]

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[26]

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[27]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83.  doi: 10.1007/BF01389295.  Google Scholar

[28]

F. Monard, Numerical implementation of geodesic X-ray transforms and their inversion, SIAM J. Imaging Sciences, 7 (2014), 1335-1357.  doi: 10.1137/130938657.  Google Scholar

[29]

F. MonardP. Stefanov and G. Uhlmann, The geodesic ray transform on Riemannian surfaces with conjugate points, Communications in Mathematical Physics, 337 (2015), 1491-1513.  doi: 10.1007/s00220-015-2328-6.  Google Scholar

[30]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.   Google Scholar

[31]

R. G. Mukhometov, On a problem of reconstructing Riemannian metrics, Siberian Math. J., 22 (1981), 119-135.   Google Scholar

[32]

R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an n-dimensional space (Russian), Dokl. Akad. Nauk SSSR, 243 (1978), 41-44.   Google Scholar

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G. PaternainM. Salo and G. Uhlmann, Tensor tomography on simple surfaces, Invent. Math., 193 (2013), 229-247.  doi: 10.1007/s00222-012-0432-1.  Google Scholar

[34]

G. PaternainM. Salo and G. Uhlmann, Tensor tomography: Progress and challenges, Chinese Annals of Math. Ser. B, 35 (2014), 399-428.  doi: 10.1007/s11401-014-0834-z.  Google Scholar

[35]

G. P. Paternain, M. Salo, G. Uhlmann and H. Zhou, The geodesic X-ray transform with matrix weights, to appear in Amer. J. Math. Google Scholar

[36]

G. P. Paternain and H. Zhou, Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.  doi: 10.2140/apde.2016.9.1903.  Google Scholar

[37]

L. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Sibirsk. Mat. Zh., 29 (1988), 114-130.  doi: 10.1007/BF00969652.  Google Scholar

[38]

L. Pestov and G. Uhlmann, On Characterization of the Range and Inversion Formulas for the Geodesic X-ray Transform, Int. Math. Res. Not., 80 (2004), 4331-4347.  doi: 10.1155/S1073792804142116.  Google Scholar

[39]

L. Pestov and G. Uhlmann, Two dimensional simple Riemannian manifolds with boundary are boundary distance rigid, Annals of Math., 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093.  Google Scholar

[40]

A. Ranjan and H. Shah, Convexity of spheres in a manifold without conjugate points, Proc. Indian Acad. Sci. (Math. Sci.), 112 (2002), 595-599.  doi: 10.1007/BF02829692.  Google Scholar

[41]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar

[42]

V. A. Sharafutdinov, Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds, J. Geom. Anal., 17 (2007), 147-187.  doi: 10.1007/BF02922087.  Google Scholar

[43]

P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett., 5 (1998), 83-96.  doi: 10.4310/MRL.1998.v5.n1.a7.  Google Scholar

[44]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.  doi: 10.1215/S0012-7094-04-12332-2.  Google Scholar

[45]

P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.  doi: 10.1090/S0894-0347-05-00494-7.  Google Scholar

[46]

P. Stefanov and G. Uhlmann, Boundary and lens rigidity, tensor tomography and analytic microlocal analysis, in Algebraic Analysis of Differential Equations, Fetschrift in Honor of Takahiro Kawai, edited by T. Aoki, H. Majima, Y. Katei and N. Tose, pp. (2008), 275–293. doi: 10.1007/978-4-431-73240-2_23.  Google Scholar

[47]

P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math., 130 (2008), 239-268.  doi: 10.1353/ajm.2008.0003.  Google Scholar

[48]

P. Stefanov and G. Uhlmann, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom., 82 (2009), 383-409.  doi: 10.4310/jdg/1246888489.  Google Scholar

[49]

P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE, 5 (2012), 219-260.  doi: 10.2140/apde.2012.5.219.  Google Scholar

[50]

P. StefanovG. Uhlmann and A. Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc., 29 (2016), 299-332.  doi: 10.1090/jams/846.  Google Scholar

[51]

P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors, to appear in Journal d'Analyse Mathematique. Google Scholar

[52]

G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120.  doi: 10.1007/s00222-015-0631-7.  Google Scholar

[53]

G. Uhlmann and H. Zhou, Journey to the Center of the Earth, arXiv: 1604.00630. Google Scholar

[54]

J. Vargo, A proof of lens rigidity in the category of analytic metrics, Math. Research Letters, 16 (2009), 1057-1069.  doi: 10.4310/MRL.2009.v16.n6.a13.  Google Scholar

[55]

E. Wiechert and K. Zoeppritz, Über Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss, Goettingen, 4 (1907), 415-549.   Google Scholar

[56]

H. Zhou, The local magnetic ray transform of tensor fields, SIAM J. Math. Anal., 50 (2018), 1753-1778.  doi: 10.1137/16M1093963.  Google Scholar

show all references

References:
[1]

G. Ainsworth, The attenuated magnetic ray transform on surfaces, Inverse Problems and Imaging, 7 (2013), 27-46.  doi: 10.3934/ipi.2013.7.27.  Google Scholar

[2]

D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems, [Russian]Uspekhi Mat. Nauk, 22 (1967), 107–172.  Google Scholar

[3]

V. I. Arnold, Some remarks on flows of line elements and frames, Sov. Math. Dokl., 2 (1961), 562-564.   Google Scholar

[4]

Y. Assylbekov and H. Zhou, Boundary and scattering rigidity problems in the presence of a magnetic field and a potential, Inverse Problems and Imaging, 9 (2015), 935-950.  doi: 10.3934/ipi.2015.9.935.  Google Scholar

[5]

G. Bao and H. Zhang, Sensitivity analysis of an inverse problem for the wave equation with caustics, J. Amer. Math. Soc., 27 (2014), 953-981.  doi: 10.1090/S0894-0347-2014-00787-6.  Google Scholar

[6]

C. Croke, Rigidity and distance between boundary points, J. Diff. Geom., 33 (1991), 445-464.  doi: 10.4310/jdg/1214446326.  Google Scholar

[7]

C. Croke, Rigidity theorems in Riemannian geometry, in Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl., 137, Springer, New York, 2004, 47–72. doi: 10.1007/978-1-4684-9375-7_4.  Google Scholar

[8]

C. Croke, Boundary and lens rigidity of finite quotients, Proc. Am. Math. Soc., 133 (2005), 3663-3668.  doi: 10.1090/S0002-9939-05-07927-X.  Google Scholar

[9]

C. Croke, Scattering rigidity with trapped geodesics, Ergodic Theory Dynam. Systems, 34 (2014), 826-836.  doi: 10.1017/etds.2012.164.  Google Scholar

[10]

C. Croke and B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector field, J. Diff. Geom., 39 (1994), 659-680.  doi: 10.4310/jdg/1214455076.  Google Scholar

[11]

N. S. Dairbekov, Integral geometry problem for nontrapping manifolds, Inverse Problems, 22 (2006), 431-445.  doi: 10.1088/0266-5611/22/2/003.  Google Scholar

[12]

N. S. DairbekovG. P. PaternainP. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609.  doi: 10.1016/j.aim.2007.05.014.  Google Scholar

[13]

N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Problems and Imaging, 4 (2010), 397-409.  doi: 10.3934/ipi.2010.4.397.  Google Scholar

[14]

J. H. Eschenburg, Local convexity and nonnegative curvature- Gromov proof of the sphere theorem, Invent. Math., 84 (1986), 507-522.  doi: 10.1007/BF01388744.  Google Scholar

[15]

B. FrigyikP. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108.  doi: 10.1007/s12220-007-9007-6.  Google Scholar

[16]

R. E. Greene and H. Wu, C convex functions and manifolds of positive curvature, Acta Math., 137 (1976), 209-245.  doi: 10.1007/BF02392418.  Google Scholar

[17]

C. Guillarmou, Lens rigidity for manifolds with hyperbolic trapped set, J. Amer. Math. Soc., 30 (2017), 561-599.  doi: 10.1090/jams/865.  Google Scholar

[18]

G. Herglotz, Über die Elastizität der Erde bei Berücksichtigung ihrer variablen Dichte, Zeitschr. für Math. Phys., 52 (1905), 275-299.   Google Scholar

[19]

P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528.  doi: 10.4310/CAG.2012.v20.n3.a3.  Google Scholar

[20]

P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664.  doi: 10.1007/s12220-010-9162-z.  Google Scholar

[21]

S. Holman, Generic local uniqueness and stability in polarization tomography, J. Geom. Anal., 23 (2013), 229-269.  doi: 10.1007/s12220-011-9245-5.  Google Scholar

[22]

S. Holman and P. Stefanov, The weighted Doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130.  doi: 10.3934/ipi.2010.4.111.  Google Scholar

[23]

V. Krishnan, A support theorem for the geodesic ray transform on functions, J. Fourier Anal. Appl., 15 (2009), 515-520.  doi: 10.1007/s00041-009-9061-5.  Google Scholar

[24]

V. Krishnan and P. Stefanov, A support theorem for the geodesic ray transform of symmetric tensor fields, Inverse Problems and Imaging, 3 (2009), 453-464.  doi: 10.3934/ipi.2009.3.453.  Google Scholar

[25]

M. LassasV. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., 325 (2003), 767-793.  doi: 10.1007/s00208-002-0407-4.  Google Scholar

[26]

R. B. Melrose, Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean Spaces, Marcel Dekker, 1994.  Google Scholar

[27]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83.  doi: 10.1007/BF01389295.  Google Scholar

[28]

F. Monard, Numerical implementation of geodesic X-ray transforms and their inversion, SIAM J. Imaging Sciences, 7 (2014), 1335-1357.  doi: 10.1137/130938657.  Google Scholar

[29]

F. MonardP. Stefanov and G. Uhlmann, The geodesic ray transform on Riemannian surfaces with conjugate points, Communications in Mathematical Physics, 337 (2015), 1491-1513.  doi: 10.1007/s00220-015-2328-6.  Google Scholar

[30]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.   Google Scholar

[31]

R. G. Mukhometov, On a problem of reconstructing Riemannian metrics, Siberian Math. J., 22 (1981), 119-135.   Google Scholar

[32]

R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an n-dimensional space (Russian), Dokl. Akad. Nauk SSSR, 243 (1978), 41-44.   Google Scholar

[33]

G. PaternainM. Salo and G. Uhlmann, Tensor tomography on simple surfaces, Invent. Math., 193 (2013), 229-247.  doi: 10.1007/s00222-012-0432-1.  Google Scholar

[34]

G. PaternainM. Salo and G. Uhlmann, Tensor tomography: Progress and challenges, Chinese Annals of Math. Ser. B, 35 (2014), 399-428.  doi: 10.1007/s11401-014-0834-z.  Google Scholar

[35]

G. P. Paternain, M. Salo, G. Uhlmann and H. Zhou, The geodesic X-ray transform with matrix weights, to appear in Amer. J. Math. Google Scholar

[36]

G. P. Paternain and H. Zhou, Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.  doi: 10.2140/apde.2016.9.1903.  Google Scholar

[37]

L. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Sibirsk. Mat. Zh., 29 (1988), 114-130.  doi: 10.1007/BF00969652.  Google Scholar

[38]

L. Pestov and G. Uhlmann, On Characterization of the Range and Inversion Formulas for the Geodesic X-ray Transform, Int. Math. Res. Not., 80 (2004), 4331-4347.  doi: 10.1155/S1073792804142116.  Google Scholar

[39]

L. Pestov and G. Uhlmann, Two dimensional simple Riemannian manifolds with boundary are boundary distance rigid, Annals of Math., 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093.  Google Scholar

[40]

A. Ranjan and H. Shah, Convexity of spheres in a manifold without conjugate points, Proc. Indian Acad. Sci. (Math. Sci.), 112 (2002), 595-599.  doi: 10.1007/BF02829692.  Google Scholar

[41]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar

[42]

V. A. Sharafutdinov, Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds, J. Geom. Anal., 17 (2007), 147-187.  doi: 10.1007/BF02922087.  Google Scholar

[43]

P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett., 5 (1998), 83-96.  doi: 10.4310/MRL.1998.v5.n1.a7.  Google Scholar

[44]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.  doi: 10.1215/S0012-7094-04-12332-2.  Google Scholar

[45]

P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.  doi: 10.1090/S0894-0347-05-00494-7.  Google Scholar

[46]

P. Stefanov and G. Uhlmann, Boundary and lens rigidity, tensor tomography and analytic microlocal analysis, in Algebraic Analysis of Differential Equations, Fetschrift in Honor of Takahiro Kawai, edited by T. Aoki, H. Majima, Y. Katei and N. Tose, pp. (2008), 275–293. doi: 10.1007/978-4-431-73240-2_23.  Google Scholar

[47]

P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math., 130 (2008), 239-268.  doi: 10.1353/ajm.2008.0003.  Google Scholar

[48]

P. Stefanov and G. Uhlmann, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom., 82 (2009), 383-409.  doi: 10.4310/jdg/1246888489.  Google Scholar

[49]

P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE, 5 (2012), 219-260.  doi: 10.2140/apde.2012.5.219.  Google Scholar

[50]

P. StefanovG. Uhlmann and A. Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc., 29 (2016), 299-332.  doi: 10.1090/jams/846.  Google Scholar

[51]

P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors, to appear in Journal d'Analyse Mathematique. Google Scholar

[52]

G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120.  doi: 10.1007/s00222-015-0631-7.  Google Scholar

[53]

G. Uhlmann and H. Zhou, Journey to the Center of the Earth, arXiv: 1604.00630. Google Scholar

[54]

J. Vargo, A proof of lens rigidity in the category of analytic metrics, Math. Research Letters, 16 (2009), 1057-1069.  doi: 10.4310/MRL.2009.v16.n6.a13.  Google Scholar

[55]

E. Wiechert and K. Zoeppritz, Über Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss, Goettingen, 4 (1907), 415-549.   Google Scholar

[56]

H. Zhou, The local magnetic ray transform of tensor fields, SIAM J. Math. Anal., 50 (2018), 1753-1778.  doi: 10.1137/16M1093963.  Google Scholar

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