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]

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

[2]

Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 657-709, 896.  Google Scholar

[3]

Tôhoku Math. J., 35 (1983), 155-172. doi: 10.2748/tmj/1178229047.  Google Scholar

[4]

Bull. Soc. Math. France, 83 (1955), 279-330.  Google Scholar

[5]

Internat. J. Math., 17 (2006), 641-664. doi: 10.1142/S0129167X06003655.  Google Scholar

[6]

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]

Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.  Google Scholar

[8]

Math. Ann., 336 (2006), 311-334. doi: 10.1007/s00208-006-0004-z.  Google Scholar

[9]

Differential Geom. Appl., 26 (2008), 244-266.  Google Scholar

[10]

Inverse Probl. Imaging, 3 (2009), 1-15. doi: 10.3934/ipi.2009.3.1.  Google Scholar

[11]

Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.  Google Scholar

[12]

Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.  Google Scholar

[13]

Math. Ann., 197 (1972), 323-334. doi: 10.1007/BF01428204.  Google Scholar

[14]

Comm. Anal. Geom., 11 (2003), 207-221.  Google Scholar

[15]

Ann. Sci. École Norm. Sup. (4), 34 (2001), 771-787.  Google Scholar

[16]

Comm. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804.  Google Scholar

[17]

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]

J. Math. Sci. (N. Y.), 146 (2007), 6313-6390. doi: 10.1007/s10958-007-0472-z.  Google Scholar

[19]

The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963.  Google Scholar

[20]

Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar

[21]

London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.  Google Scholar

[22]

Habilitation, Ludwig Maximilians University in Munchen, 2001. Google Scholar

[23]

Ann. of Math. (2), 121 (1985), 169-186. doi: 10.2307/1971195.  Google Scholar

[24]

Trans. Amer. Math. Soc., 358 (2006), 2415-2423. doi: 10.1090/S0002-9947-06-04090-6.  Google Scholar

[25]

Amer. Math. J., 98 (1976), 1059-1078. doi: 10.2307/2374041.  Google Scholar

[26]

Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

show all references

References:
[1]

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

[2]

Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 657-709, 896.  Google Scholar

[3]

Tôhoku Math. J., 35 (1983), 155-172. doi: 10.2748/tmj/1178229047.  Google Scholar

[4]

Bull. Soc. Math. France, 83 (1955), 279-330.  Google Scholar

[5]

Internat. J. Math., 17 (2006), 641-664. doi: 10.1142/S0129167X06003655.  Google Scholar

[6]

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]

Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.  Google Scholar

[8]

Math. Ann., 336 (2006), 311-334. doi: 10.1007/s00208-006-0004-z.  Google Scholar

[9]

Differential Geom. Appl., 26 (2008), 244-266.  Google Scholar

[10]

Inverse Probl. Imaging, 3 (2009), 1-15. doi: 10.3934/ipi.2009.3.1.  Google Scholar

[11]

Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.  Google Scholar

[12]

Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.  Google Scholar

[13]

Math. Ann., 197 (1972), 323-334. doi: 10.1007/BF01428204.  Google Scholar

[14]

Comm. Anal. Geom., 11 (2003), 207-221.  Google Scholar

[15]

Ann. Sci. École Norm. Sup. (4), 34 (2001), 771-787.  Google Scholar

[16]

Comm. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804.  Google Scholar

[17]

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]

J. Math. Sci. (N. Y.), 146 (2007), 6313-6390. doi: 10.1007/s10958-007-0472-z.  Google Scholar

[19]

The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963.  Google Scholar

[20]

Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar

[21]

London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.  Google Scholar

[22]

Habilitation, Ludwig Maximilians University in Munchen, 2001. Google Scholar

[23]

Ann. of Math. (2), 121 (1985), 169-186. doi: 10.2307/1971195.  Google Scholar

[24]

Trans. Amer. Math. Soc., 358 (2006), 2415-2423. doi: 10.1090/S0002-9947-06-04090-6.  Google Scholar

[25]

Amer. Math. J., 98 (1976), 1059-1078. doi: 10.2307/2374041.  Google Scholar

[26]

Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

[1]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[2]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313

[3]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021063

[4]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[5]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[6]

Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021070

[7]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002

[8]

Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011

[9]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019

[10]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909

[11]

Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021046

[12]

Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047

[13]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015

[14]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021058

[15]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011

[16]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404

[17]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021077

[18]

Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220

[19]

Jun Tu, Zijiao Sun, Min Huang. Supply chain coordination considering e-tailer's promotion effort and logistics provider's service effort. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021062

[20]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]