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, (1970), 1.   Google Scholar

[2]

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

[5]

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

[6]

A.-P. Calderón, On an inverse boundary value problem,, in, (1980), 65.   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.  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.  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.   Google Scholar

[10]

C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds,, Inverse Probl. Imaging, 3 (2009), 1.  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.  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.  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.  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.   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.   Google Scholar

[16]

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

[17]

J. Lelong-Ferrand, Geometrical interpretation of scalar curvature and regularity of conformal homeomorphisms,, in, (1976), 91.   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.  doi: 10.1007/s10958-007-0472-z.  Google Scholar

[19]

H. L. Royden, "Real Analysis,", The Macmillan Co., (1963).   Google Scholar

[20]

A. Sard, The measure of the critical values of differentiable maps,, Bull. Amer. Math. Soc., 48 (1942), 883.  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 (1989).   Google Scholar

[22]

U. Semmelmann, "Conformal Killing Forms on Riemannian Manifolds,", Habilitation, (2001).   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

show all references

References:
[1]

R. Abraham, Bumpy metrics,, in, (1970), 1.   Google Scholar

[2]

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

[5]

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

[6]

A.-P. Calderón, On an inverse boundary value problem,, in, (1980), 65.   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.  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.  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.   Google Scholar

[10]

C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds,, Inverse Probl. Imaging, 3 (2009), 1.  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.  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.  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.  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.   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.   Google Scholar

[16]

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

[17]

J. Lelong-Ferrand, Geometrical interpretation of scalar curvature and regularity of conformal homeomorphisms,, in, (1976), 91.   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.  doi: 10.1007/s10958-007-0472-z.  Google Scholar

[19]

H. L. Royden, "Real Analysis,", The Macmillan Co., (1963).   Google Scholar

[20]

A. Sard, The measure of the critical values of differentiable maps,, Bull. Amer. Math. Soc., 48 (1942), 883.  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 (1989).   Google Scholar

[22]

U. Semmelmann, "Conformal Killing Forms on Riemannian Manifolds,", Habilitation, (2001).   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

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