Abstract
Consider a fractional operator \(P^s\), \(0<s<1\), for connection Laplacian \(P:=\nabla ^*\nabla +A\) on a smooth Hermitian vector bundle over a closed, connected Riemannian manifold of dimension \(n\ge 2\). We show that local knowledge of the metric, Hermitian bundle, connection, potential, and source-to-solution map associated with \(P^s\) determines these structures globally. This extends a result known for the fractional Laplace-Beltrami operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albin, P., Guillarmou, C., Tzou, L., Uhlmann, G.: Inverse boundary problems for systems in two dimensions. Ann. Henri Poincaré. 14(6), 1551–1571 (2013)
Bao, N., Cao, C., Fischetti, S., Keeler, C.: Towards bulk metric reconstruction from extremal area variations. Class. Quant. Gravity 36(18), 185002 (2019)
Bhattacharyya, S., Ghosh, T., Uhlmann, G.: Inverse problems for the fractional-Laplacian with lower order non-local perturbations. Trans. Am. Math. Soc. 374(5), 3053–3075 (2021)
Cekić, M., Lin, Y.-H., Rüland, A.: The Calderón problem for the fractional Schrödinger equation with drift. Cal. Var. Part. Differ. Equ. 59(3), 1–46 (2020)
Chang, S.-Y.A., del Mar Gonzalez, M.: Fractional Laplacian in conformal geometry. Adv. Math. 226(2), 1410–1432 (2011)
Covi, G., Mönkkönen, K., Railo, J., Uhlmann, G.: The higher order fractional Calderón problem for linear local operators: Uniqueness. Adv. Math. 399, 108246 (2022)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Differ. Equ. 32(8), 1245–1260 (2007)
del Mar González Nogueras, M., Qing, J.: Fractional conformal Laplacians and fractional Yamabe problems. Anal. PDE 6(7), 1535–1576 (2013)
Eller, M., Toundykov, D.: A global Holmgren theorem for multidimensional hyperbolic partial differential equations. Appl. Anal. 91(1), 69–90 (2012)
Eller, M., Isakov, V., Nakamura, G., Tataru, D.: Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems. In: Nonlinear partial differential equations and their applications: Collège de France seminar. Volume XIV. Elsevier. pp. 329–349 (2002)
Feizmohammadi, A., Ghosh, T., Krupchyk, K., Uhlmann, G.: Fractional anisotropic Calderón problem on closed Riemannian manifolds. (2021). arXiv:2112.03480
Gilkey, P.B.: Invariance theory: the heat equation and the Atiyah-Singer index theorem, vol. 16. CRC Press, Boca Raton (2018)
Ghosh, T., Lin, Y.-H., Xiao, J.: The Calderón problem for variable coefficients nonlocal elliptic operators. Commun. Part. Differ. Equ. 42(12), 1923–1961 (2017)
Grubb, G.: Distributions and operators, vol. 252. Springer, Berlin (2008)
Ghosh, T., Salo, M., Uhlmann, G.: The Calderón problem for the fractional Schrödinger equation. Anal. PDE 13(2), 455–475 (2020)
Helin, T., Lassas, M., Oksanen, L., Saksala, T.: Correlation based passive imaging with a white noise source. J. Math. Pures Appl. 116, 132–160 (2018)
Kachalov, A., Kurylev, Y., Lassas, M.: Inverse boundary spectral problems. Chapman and Hall/CRC, Boca Raton (2001)
Kurylev, Y., Oksanen, L., Paternain, G.P.: Inverse problems for the connection Laplacian. J. Differ. Geom. 110(3), 457–494 (2018)
Lee, John M.: Introduction to Riemannian manifolds. Springer, (2018)
Lin, F.H.: A uniqueness theorem for parabolic equations. Commun. Pure Appl. Math. 43(1), 127–136 (1990)
Ludewig, M.: Strong short-time asymptotics and convolution approximation of the heat kernel. Ann. Glob. Anal. Geom. 55(2), 371–394 (2019)
Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. 1, 1093–1110 (2005)
Quan, H., Uhlmann, G.: The Calderón problem for the fractional Dirac operator. (2022). . arXiv:2204.00965
Rüland, A., Salo, M.: The fractional Calderón problem: low regularity and stability. Nonlinear Anal. 193, 111529 (2020)
Taylor, M.E.: Partial differential equations 1, Basic theory. Springer, Berlin (1996)
Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Problems 25(12), 123011 (2009)
Acknowledgements
The author would like to thank Gunther Uhlmann for suggesting this problem and his support throughout; and Hadrian Quan for helpful references.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no relevant financial or non-financial interests to declare.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chien, CK.K. An Inverse Problem for Fractional Connection Laplacians. J Geom Anal 33, 375 (2023). https://doi.org/10.1007/s12220-023-01426-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01426-3