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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[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.

[25]

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

[26]

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

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[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.

[25]

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

[26]

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

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