August  2012, 6(3): 523-530. doi: 10.3934/ipi.2012.6.523

Nowhere conformally homogeneous manifolds and limiting Carleman weights

1. 

Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100 FI-00076 Aalto, Finland

2. 

Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, P.O. Box 35 FI-40014 Jyväskylä, Finland

Received  September 2010 Revised  July 2012 Published  September 2012

In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analogue of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symmetries. Consequently, generic manifolds of dimension $\geq 3$ do not admit nontrivial conformal Killing vector fields near any point. As an application to the inverse problem of Calderón on manifolds, this implies that generic manifolds of dimension $\geq 3$ do not admit limiting Carleman weights near any point.
Citation: Tony Liimatainen, Mikko Salo. Nowhere conformally homogeneous manifolds and limiting Carleman weights. Inverse Problems & Imaging, 2012, 6 (3) : 523-530. doi: 10.3934/ipi.2012.6.523
References:
[1]

R. Abraham, Bumpy metrics, in "Global Analysis" (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, 1968), AMS, Providence, RI, (1970), 1-3.  Google Scholar

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D. V. Anosov, Generic properties of closed geodesics, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 657-709, 896.  Google Scholar

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S. Bando and H. Urakawa, Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tôhoku Math. J., 35 (1983), 155-172. doi: 10.2748/tmj/1178229047.  Google Scholar

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M. Berger, Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France, 83 (1955), 279-330.  Google Scholar

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T. Branson, A. Čap, M. Eastwood and A. R. Gover, Prolongations of geometric overdetermined systems, Internat. J. Math., 17 (2006), 641-664. doi: 10.1142/S0129167X06003655.  Google Scholar

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A.-P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Jeneiro, 1980), Soc. Brasileira de Matemática, Rio de Janeiro, (1980), 65-73.  Google Scholar

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D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.  Google Scholar

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A. R. Gover, Laplacian operators and Q-curvature on conformally Einstein manifolds, Math. Ann., 336 (2006), 311-334. doi: 10.1007/s00208-006-0004-z.  Google Scholar

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A. R. Gover and J. Šilhan, The conformal Killing equation on forms -prolongations and applications, Differential Geom. Appl., 26 (2008), 244-266.  Google Scholar

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C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds, Inverse Probl. Imaging, 3 (2009), 1-15. doi: 10.3934/ipi.2009.3.1.  Google Scholar

[11]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.  Google Scholar

[12]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.  Google Scholar

[13]

W. Klingenberg and F. Takens, Generic properties of geodesic flows, Math. Ann., 197 (1972), 323-334. doi: 10.1007/BF01428204.  Google Scholar

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M. Lassas, M. Taylor and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom., 11 (2003), 207-221.  Google Scholar

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M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. École Norm. Sup. (4), 34 (2001), 771-787.  Google Scholar

[16]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804.  Google Scholar

[17]

J. Lelong-Ferrand, Geometrical interpretation of scalar curvature and regularity of conformal homeomorphisms, in "Differential Geometry and Relativity" (eds. M. Caheu and M. Flato), Mathematical Phys. and Appl. Math., Vol. 3, Reidel, Dordrecht, (1976), 91-105.  Google Scholar

[18]

Yu. G. Nikonorov, E. D. Rodionov and V. V. Slavskiĭ, Geometry of homogeneous Riemannian manifolds, J. Math. Sci. (N. Y.), 146 (2007), 6313-6390. doi: 10.1007/s10958-007-0472-z.  Google Scholar

[19]

H. L. Royden, "Real Analysis," The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963.  Google Scholar

[20]

A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar

[21]

D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.  Google Scholar

[22]

U. Semmelmann, "Conformal Killing Forms on Riemannian Manifolds," Habilitation, Ludwig Maximilians University in Munchen, 2001. Google Scholar

[23]

T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2), 121 (1985), 169-186. doi: 10.2307/1971195.  Google Scholar

[24]

M. E. Taylor, Existence and regularity of isometries, Trans. Amer. Math. Soc., 358 (2006), 2415-2423. doi: 10.1090/S0002-9947-06-04090-6.  Google Scholar

[25]

K. Uhlenbeck, Generic properties of eigenfunctions, Amer. Math. J., 98 (1976), 1059-1078. doi: 10.2307/2374041.  Google Scholar

[26]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

show all references

References:
[1]

R. Abraham, Bumpy metrics, in "Global Analysis" (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, 1968), AMS, Providence, RI, (1970), 1-3.  Google Scholar

[2]

D. V. Anosov, Generic properties of closed geodesics, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 657-709, 896.  Google Scholar

[3]

S. Bando and H. Urakawa, Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tôhoku Math. J., 35 (1983), 155-172. doi: 10.2748/tmj/1178229047.  Google Scholar

[4]

M. Berger, Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France, 83 (1955), 279-330.  Google Scholar

[5]

T. Branson, A. Čap, M. Eastwood and A. R. Gover, Prolongations of geometric overdetermined systems, Internat. J. Math., 17 (2006), 641-664. doi: 10.1142/S0129167X06003655.  Google Scholar

[6]

A.-P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Jeneiro, 1980), Soc. Brasileira de Matemática, Rio de Janeiro, (1980), 65-73.  Google Scholar

[7]

D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.  Google Scholar

[8]

A. R. Gover, Laplacian operators and Q-curvature on conformally Einstein manifolds, Math. Ann., 336 (2006), 311-334. doi: 10.1007/s00208-006-0004-z.  Google Scholar

[9]

A. R. Gover and J. Šilhan, The conformal Killing equation on forms -prolongations and applications, Differential Geom. Appl., 26 (2008), 244-266.  Google Scholar

[10]

C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds, Inverse Probl. Imaging, 3 (2009), 1-15. doi: 10.3934/ipi.2009.3.1.  Google Scholar

[11]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.  Google Scholar

[12]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.  Google Scholar

[13]

W. Klingenberg and F. Takens, Generic properties of geodesic flows, Math. Ann., 197 (1972), 323-334. doi: 10.1007/BF01428204.  Google Scholar

[14]

M. Lassas, M. Taylor and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom., 11 (2003), 207-221.  Google Scholar

[15]

M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. École Norm. Sup. (4), 34 (2001), 771-787.  Google Scholar

[16]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804.  Google Scholar

[17]

J. Lelong-Ferrand, Geometrical interpretation of scalar curvature and regularity of conformal homeomorphisms, in "Differential Geometry and Relativity" (eds. M. Caheu and M. Flato), Mathematical Phys. and Appl. Math., Vol. 3, Reidel, Dordrecht, (1976), 91-105.  Google Scholar

[18]

Yu. G. Nikonorov, E. D. Rodionov and V. V. Slavskiĭ, Geometry of homogeneous Riemannian manifolds, J. Math. Sci. (N. Y.), 146 (2007), 6313-6390. doi: 10.1007/s10958-007-0472-z.  Google Scholar

[19]

H. L. Royden, "Real Analysis," The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963.  Google Scholar

[20]

A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar

[21]

D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.  Google Scholar

[22]

U. Semmelmann, "Conformal Killing Forms on Riemannian Manifolds," Habilitation, Ludwig Maximilians University in Munchen, 2001. Google Scholar

[23]

T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2), 121 (1985), 169-186. doi: 10.2307/1971195.  Google Scholar

[24]

M. E. Taylor, Existence and regularity of isometries, Trans. Amer. Math. Soc., 358 (2006), 2415-2423. doi: 10.1090/S0002-9947-06-04090-6.  Google Scholar

[25]

K. Uhlenbeck, Generic properties of eigenfunctions, Amer. Math. J., 98 (1976), 1059-1078. doi: 10.2307/2374041.  Google Scholar

[26]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

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