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:
[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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[53]

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

[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.

[55]

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

[56]

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

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[53]

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

[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.

[55]

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

[56]

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

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