doi: 10.3934/ipi.2022028
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Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary

Department of Mathematics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27607, USA

*Corresponding author: Teemu Saksala

Received  January 2022 Revised  April 2022 Early access May 2022

Fund Project: EP was supported by the AWM and NSF-DMS grant # 1953892 for presenting this paper at the 2022 Joint Mathematics Meetings

In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the manifold, the respective distance function to the points on the measurement region is known. This geometric inverse problem has many connections to seismology, in particular to microseismicity. The reconstruction is based on embedding the manifold in a function space. This requires the differentiation of the distance functions. Therefore this paper also studies some global regularity properties of the distance function on a compact Riemannian manifold with strictly convex boundary.

Citation: Ella Pavlechko, Teemu Saksala. Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary. Inverse Problems and Imaging, doi: 10.3934/ipi.2022028
References:
[1]

R. Alexander and S. Alexander, Geodesics in Riemannian manifolds-with-boundary, Indiana University Mathematics Journal, 30 (1981), 481-488.  doi: 10.1512/iumj.1981.30.30039.

[2]

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

[3]

R. BartoloE. CaponioA. V. Germinario and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calculus of Variations and Partial Differential Equations, 40 (2011), 335-356.  doi: 10.1007/s00526-010-0343-1.

[4]

M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (Bc–Method), Communications in Partial Differential Equations, 17 (1992), 767-804.  doi: 10.1080/03605309208820863.

[5]

R. L. Bishop, Infinitesimal convexity implies local convexity, Indiana Univ. Math. J., 24 (1974/75), 169-172.  doi: 10.1512/iumj.1975.24.24014.

[6]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, volume 33., American Mathematical Soc., 2001. doi: 10.1090/gsm/033.

[7]

D. Burago and S. Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, Annals of Mathematics, 171 (2010), 1183-1211.  doi: 10.4007/annals.2010.171.1183.

[8] V. Cerveny, Seismic Ray Theory, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511529399.
[9]

C. B. Croke, Rigidity and the distance between boundary points, Journal of Differential Geometry, 33 (1991), 445-464.  doi: 10.4310/jdg/1214446326.

[10]

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

[11]

M. V. de Hoop, J. Ilmavirta, M. Lassas and T. Saksala, Determination of a compact Finsler manifold from its boundary distance map and an inverse problem in elasticity, (to appear) Communications in Analysis and Geometry, arXiv preprint, arXiv: 1901.03902, 2019.

[12]

M. V. de Hoop, J. Ilmavirta, M. Lassas and T. Saksala, Stable reconstruction of simple Riemannian manifolds from unknown interior sources, arXiv preprint, arXiv: 2102.11799, 2021.

[13]

M. V. de HoopJ. IlmavirtaM. Lassas and T. Saksala, A foliated and reversible Finsler manifold is determined by its broken scattering relation, Pure and Applied Analysis, 3 (2021), 789-811.  doi: 10.2140/paa.2021.3.789.

[14]

M. V. de Hoop and T. Saksala, Inverse problem of travel time difference functions on a compact Riemannian manifold with boundary, The Journal of Geometric Analysis, 29 (2019), 3308-3327.  doi: 10.1007/s12220-018-00111-0.

[15] M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. 
[16]

J. J. Duistermaat and L. Hörmander, Fourier integral operators. Ⅱ, Acta Mathematica, 128 (1972), 183-269.  doi: 10.1007/BF02392165.

[17]

J. J. Duistermaat and L. Hörmander, Fourier Integral Operators, volume 2., Springer, 1996.

[18]

A. Greenleaf and G. Uhlmann, Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.  doi: 10.1007/BF02096882.

[19]

J.-I. Itoh and M. Tanaka, The dimension of a cut locus on a smooth Riemannian manifold, Tohoku Mathematical Journal, Second Series, 50 (1998), 571-575.  doi: 10.2748/tmj/1178224899.

[20]

S. Ivanov, Local monotonicity of Riemannian and Finsler volume with respect to boundary distances, Geometriae Dedicata, 164 (2013), 83-96.  doi: 10.1007/s10711-012-9760-y.

[21]

S. Ivanov, Distance difference representations of Riemannian manifolds, Geometriae Dedicata, 207 (2020), 167-192.  doi: 10.1007/s10711-019-00491-9.

[22]

S. Ivanov, Distance difference functions on non-convex boundaries of Riemannian manifolds, 2022.

[23]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, volume 123 of Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[24]

A. KatsudaY. Kurylev and M. Lassas, Stability of boundary distance representation and reconstruction of Riemannian manifolds, Inverse Problems & Imaging, 1 (2007), 135-157.  doi: 10.3934/ipi.2007.1.135.

[25]

W. Klingenberg, Riemannian Geometry, volume 1., Walter de Gruyter, 1982.

[26]

Y. Kurylev, Multidimensional Gel'fand inverse problem and boundary distance map, Inverse Problems Related with Geometry (ed. H. Soga), (1997), pages 1–15.

[27]

Y. KurylevM. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, American Journal of Mathematics, 132 (2010), 529-562.  doi: 10.1353/ajm.0.0103.

[28]

Y. KurylevM. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Inventiones Mathematicae, 212 (2018), 781-857.  doi: 10.1007/s00222-017-0780-y.

[29]

Y. KurylevL. Oksanen and G. P. Paternain, Inverse problems for the connection Laplacian, Journal of Differential Geometry, 110 (2018), 457-494.  doi: 10.4310/jdg/1542423627.

[30]

M. Lassas, Inverse problems for linear and non-linear hyperbolic equations, Proceedings of International Congress of Mathematicians – 2018 Rio de Janeiro, 3 (2018), 3739-3760. 

[31]

M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Mathematical Journal, 163 (2014), 1071-1103.  doi: 10.1215/00127094-2649534.

[32]

M. Lassas and T. Saksala, Determination of a Riemannian manifold from the distance difference functions, Asian Journal of Mathematics, 23 (2019), 173-200.  doi: 10.4310/AJM.2019.v23.n2.a1.

[33]

M. LassasT. Saksala and H. Zhou, Reconstruction of a compact manifold from the scattering data of internal sources, Inverse Problems & Imaging, 12 (2018), 993-1031.  doi: 10.3934/ipi.2018042.

[34]

M. LassasG. Uhlmann and Y. Wang, Inverse problems for semilinear wave equations on Lorentzian manifolds, Communications in Mathematical Physics, 360 (2018), 555-609.  doi: 10.1007/s00220-018-3135-7.

[35] J. M. Lee, Introduction to Smooth Manifolds, Springer, 2013. 
[36] J. M. Lee, Introduction to Riemannian Manifolds, Springer, 2018. 
[37] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge university press, 1995.  doi: 10.1017/CBO9780511623813.
[38]

R. Meyerson, Stitching data: Recovering a manifold's geometry from geodesic intersections, The Journal of Geometric Analysis, 32 (2022), Paper No. 95, 22 pp. doi: 10.1007/s12220-021-00815-w.

[39]

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

[40]

T. Milne and A.-R. Mansouri, Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation, Trans. Amer. Math. Soc., 371 (2019), 8781-8810.  doi: 10.1090/tran/7630.

[41]

K. Mönkkönen, Boundary rigidity for Randers metrics, arXiv preprint arXiv: 2010.11484, 2020.

[42]

V. Ozols, Cut loci in Riemannian manifolds, Tohoku Mathematical Journal, Second Series, 26 (1974), 219-227.  doi: 10.2748/tmj/1178241180.

[43]

G. P. Paternain, Geodesic Flows, volume 180., Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-1600-1.

[44]

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

[45]

P. Petersen, Riemannian Geometry, volume 171., Springer, 2006.

[46]

A. Sard, Hausdorff measure of critical images on Banach manifolds, American Journal of Mathematics, 87 (1965), 158-174.  doi: 10.2307/2373229.

[47]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, volume 1, Walter de Gruyter, 1994. doi: 10.1515/9783110900095.

[48]

P. StefanovG. Uhlmann and A. Vasy, Boundary rigidity with partial data, Journal of the American Mathematical Society, 29 (2016), 299-332.  doi: 10.1090/jams/846.

[49]

P. StefanovG. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic x-ray transform in the normal gauge, Annals of Mathematics, 194 (2021), 1-95.  doi: 10.4007/annals.2021.194.1.1.

[50]

G. Uhlmann, Inverse boundary value problems for partial differential equations, In Proceedings of the International Congress of Mathematicians, Berlin, 1998, pages 77–86, .

[51]

Y. Wang and T. Zhou, Inverse problems for quadratic derivative nonlinear wave equations, Communications in Partial Differential Equations, 44 (2019), 1140-1158.  doi: 10.1080/03605302.2019.1612908.

[52]

H. Whitney, Complex Analytic Varieties, volume 131., Addison-Wesley Reading, 1972.

show all references

References:
[1]

R. Alexander and S. Alexander, Geodesics in Riemannian manifolds-with-boundary, Indiana University Mathematics Journal, 30 (1981), 481-488.  doi: 10.1512/iumj.1981.30.30039.

[2]

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

[3]

R. BartoloE. CaponioA. V. Germinario and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calculus of Variations and Partial Differential Equations, 40 (2011), 335-356.  doi: 10.1007/s00526-010-0343-1.

[4]

M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (Bc–Method), Communications in Partial Differential Equations, 17 (1992), 767-804.  doi: 10.1080/03605309208820863.

[5]

R. L. Bishop, Infinitesimal convexity implies local convexity, Indiana Univ. Math. J., 24 (1974/75), 169-172.  doi: 10.1512/iumj.1975.24.24014.

[6]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, volume 33., American Mathematical Soc., 2001. doi: 10.1090/gsm/033.

[7]

D. Burago and S. Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, Annals of Mathematics, 171 (2010), 1183-1211.  doi: 10.4007/annals.2010.171.1183.

[8] V. Cerveny, Seismic Ray Theory, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511529399.
[9]

C. B. Croke, Rigidity and the distance between boundary points, Journal of Differential Geometry, 33 (1991), 445-464.  doi: 10.4310/jdg/1214446326.

[10]

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

[11]

M. V. de Hoop, J. Ilmavirta, M. Lassas and T. Saksala, Determination of a compact Finsler manifold from its boundary distance map and an inverse problem in elasticity, (to appear) Communications in Analysis and Geometry, arXiv preprint, arXiv: 1901.03902, 2019.

[12]

M. V. de Hoop, J. Ilmavirta, M. Lassas and T. Saksala, Stable reconstruction of simple Riemannian manifolds from unknown interior sources, arXiv preprint, arXiv: 2102.11799, 2021.

[13]

M. V. de HoopJ. IlmavirtaM. Lassas and T. Saksala, A foliated and reversible Finsler manifold is determined by its broken scattering relation, Pure and Applied Analysis, 3 (2021), 789-811.  doi: 10.2140/paa.2021.3.789.

[14]

M. V. de Hoop and T. Saksala, Inverse problem of travel time difference functions on a compact Riemannian manifold with boundary, The Journal of Geometric Analysis, 29 (2019), 3308-3327.  doi: 10.1007/s12220-018-00111-0.

[15] M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. 
[16]

J. J. Duistermaat and L. Hörmander, Fourier integral operators. Ⅱ, Acta Mathematica, 128 (1972), 183-269.  doi: 10.1007/BF02392165.

[17]

J. J. Duistermaat and L. Hörmander, Fourier Integral Operators, volume 2., Springer, 1996.

[18]

A. Greenleaf and G. Uhlmann, Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.  doi: 10.1007/BF02096882.

[19]

J.-I. Itoh and M. Tanaka, The dimension of a cut locus on a smooth Riemannian manifold, Tohoku Mathematical Journal, Second Series, 50 (1998), 571-575.  doi: 10.2748/tmj/1178224899.

[20]

S. Ivanov, Local monotonicity of Riemannian and Finsler volume with respect to boundary distances, Geometriae Dedicata, 164 (2013), 83-96.  doi: 10.1007/s10711-012-9760-y.

[21]

S. Ivanov, Distance difference representations of Riemannian manifolds, Geometriae Dedicata, 207 (2020), 167-192.  doi: 10.1007/s10711-019-00491-9.

[22]

S. Ivanov, Distance difference functions on non-convex boundaries of Riemannian manifolds, 2022.

[23]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, volume 123 of Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.

[24]

A. KatsudaY. Kurylev and M. Lassas, Stability of boundary distance representation and reconstruction of Riemannian manifolds, Inverse Problems & Imaging, 1 (2007), 135-157.  doi: 10.3934/ipi.2007.1.135.

[25]

W. Klingenberg, Riemannian Geometry, volume 1., Walter de Gruyter, 1982.

[26]

Y. Kurylev, Multidimensional Gel'fand inverse problem and boundary distance map, Inverse Problems Related with Geometry (ed. H. Soga), (1997), pages 1–15.

[27]

Y. KurylevM. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, American Journal of Mathematics, 132 (2010), 529-562.  doi: 10.1353/ajm.0.0103.

[28]

Y. KurylevM. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Inventiones Mathematicae, 212 (2018), 781-857.  doi: 10.1007/s00222-017-0780-y.

[29]

Y. KurylevL. Oksanen and G. P. Paternain, Inverse problems for the connection Laplacian, Journal of Differential Geometry, 110 (2018), 457-494.  doi: 10.4310/jdg/1542423627.

[30]

M. Lassas, Inverse problems for linear and non-linear hyperbolic equations, Proceedings of International Congress of Mathematicians – 2018 Rio de Janeiro, 3 (2018), 3739-3760. 

[31]

M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Mathematical Journal, 163 (2014), 1071-1103.  doi: 10.1215/00127094-2649534.

[32]

M. Lassas and T. Saksala, Determination of a Riemannian manifold from the distance difference functions, Asian Journal of Mathematics, 23 (2019), 173-200.  doi: 10.4310/AJM.2019.v23.n2.a1.

[33]

M. LassasT. Saksala and H. Zhou, Reconstruction of a compact manifold from the scattering data of internal sources, Inverse Problems & Imaging, 12 (2018), 993-1031.  doi: 10.3934/ipi.2018042.

[34]

M. LassasG. Uhlmann and Y. Wang, Inverse problems for semilinear wave equations on Lorentzian manifolds, Communications in Mathematical Physics, 360 (2018), 555-609.  doi: 10.1007/s00220-018-3135-7.

[35] J. M. Lee, Introduction to Smooth Manifolds, Springer, 2013. 
[36] J. M. Lee, Introduction to Riemannian Manifolds, Springer, 2018. 
[37] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge university press, 1995.  doi: 10.1017/CBO9780511623813.
[38]

R. Meyerson, Stitching data: Recovering a manifold's geometry from geodesic intersections, The Journal of Geometric Analysis, 32 (2022), Paper No. 95, 22 pp. doi: 10.1007/s12220-021-00815-w.

[39]

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

[40]

T. Milne and A.-R. Mansouri, Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation, Trans. Amer. Math. Soc., 371 (2019), 8781-8810.  doi: 10.1090/tran/7630.

[41]

K. Mönkkönen, Boundary rigidity for Randers metrics, arXiv preprint arXiv: 2010.11484, 2020.

[42]

V. Ozols, Cut loci in Riemannian manifolds, Tohoku Mathematical Journal, Second Series, 26 (1974), 219-227.  doi: 10.2748/tmj/1178241180.

[43]

G. P. Paternain, Geodesic Flows, volume 180., Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-1600-1.

[44]

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

[45]

P. Petersen, Riemannian Geometry, volume 171., Springer, 2006.

[46]

A. Sard, Hausdorff measure of critical images on Banach manifolds, American Journal of Mathematics, 87 (1965), 158-174.  doi: 10.2307/2373229.

[47]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, volume 1, Walter de Gruyter, 1994. doi: 10.1515/9783110900095.

[48]

P. StefanovG. Uhlmann and A. Vasy, Boundary rigidity with partial data, Journal of the American Mathematical Society, 29 (2016), 299-332.  doi: 10.1090/jams/846.

[49]

P. StefanovG. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic x-ray transform in the normal gauge, Annals of Mathematics, 194 (2021), 1-95.  doi: 10.4007/annals.2021.194.1.1.

[50]

G. Uhlmann, Inverse boundary value problems for partial differential equations, In Proceedings of the International Congress of Mathematicians, Berlin, 1998, pages 77–86, .

[51]

Y. Wang and T. Zhou, Inverse problems for quadratic derivative nonlinear wave equations, Communications in Partial Differential Equations, 44 (2019), 1140-1158.  doi: 10.1080/03605302.2019.1612908.

[52]

H. Whitney, Complex Analytic Varieties, volume 131., Addison-Wesley Reading, 1972.

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