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ISRN Geometry
Volume 2012 (2012), Article ID 983403, 26 pages
http://dx.doi.org/10.5402/2012/983403
Review Article

A Review on Metric Symmetries Used in Geometry and Physics

University of Windsor, Windsor, ON, Canada N9B 3P4

Received 8 November 2011; Accepted 19 December 2011

Academic Editor: C. Qu

Copyright © 2012 K. L. Duggal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Berger, Riemannian Geometry During the Second Half of the Twentieth Century, vol. 17 of Lecture Series, American Mathematical Society, Providence, RI, USA, 2000. View at Zentralblatt MATH
  2. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, NY, USA, 1983. View at Zentralblatt MATH
  3. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry, vol. 67, Marcel Dekker, New York, NY, USA, 1981, 2nd edition (with Easley, K. L.), 1996. View at Zentralblatt MATH
  4. D. N. Kupeli, Singular Semi-Riemannian Geometry, vol. 366, Kluwer Academic, Dodrecht, The Netherlands, 1996. View at Zentralblatt MATH
  5. A. Bejancu and K. L. Duggal, “Degenerated hypersurfaces of semi-Riemannian manifolds,” Buletinul Institutului Politehnic din Iaşi, vol. 37, no. 1–4, pp. 13–22, 1991. View at Zentralblatt MATH
  6. S. B. Myers and N. E. Steenrod, “The group of isometries of a Riemannian manifold,” Annals of Mathematics, vol. 40, no. 2, pp. 400–416, 1939. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. W. Killing, “Die Zusammensetzung der stetigen endlichen Transformations-gruppen,” Mathematische Annalen, vol. 31, no. 2, pp. 252–290, 1888. View at Publisher · View at Google Scholar
  8. S. Bochner, “Curvature and Betti numbers,” Annals of Mathematics, vol. 49, pp. 379–390, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Y. Watanabe, “Integral inequalities in compact orientable manifolds, Riemannian or Kählerian,” Kōdai Mathematical Seminar Reports, vol. 20, pp. 264–271, 1968. View at Publisher · View at Google Scholar
  10. K. Yano, “On harmonic and Killing vector fields,” Annals of Mathematics, vol. 55, pp. 38–45, 1952. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. K. Yano, Integral Formulas in Riemannian Geometry, Marcel Dekker, New York, NY, USA, 1970.
  12. M. Berger, “Trois remarques sur les variétés riemanniennes à courbure positive,” Comptes Rendus de l'Académie des Sciences, vol. 263, pp. A76–A78, 1966. View at Zentralblatt MATH
  13. L. P. Eisenhart, “Symmetric tensors of the second order whose first covariant derivatives are zero,” Transactions of the American Mathematical Society, vol. 25, no. 2, pp. 297–306, 1923. View at Publisher · View at Google Scholar
  14. J. Levine and G. H. Katzin, “Conformally flat spaces admitting special quadratic first integrals. I. Symmetric spaces,” Tensor, vol. 19, pp. 317–328, 1968. View at Zentralblatt MATH
  15. K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, New York, NY, USA, 1965.
  16. R. Sharma, “Second order parallel tensor in real and complex space forms,” International Journal of Mathematics and Mathematical Sciences, vol. 12, no. 4, pp. 787–790, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. K. L. Duggal and R. Sharma, Symmetries of Spacetimes and Riemannian Manifolds, vol. 487, Kluwer Academic, Dodrecht, The Netherlands, 1999.
  18. D. E. Blair, Contact Manifolds in Riemannian Geometry, vol. 509 of Lecture notes in Math, Springer, Berlin, Germany, 1976.
  19. D. E. Blair, “Two remarks on contact metric structures,” The Tohoku Mathematical Journal, vol. 29, no. 3, pp. 319–324, 1977. View at Zentralblatt MATH
  20. D. E. Blair and J. N. Patnaik, “Contact manifolds with characteristic vector field annihilated by the curvature,” Bulletin of the Institute of Mathematics, vol. 9, no. 4, pp. 533–545, 1981. View at Zentralblatt MATH
  21. R. Sharma, “Second order parallel tensors on contact manifolds,” Algebras, Groups and Geometries, vol. 7, no. 2, pp. 145–152, 1990. View at Zentralblatt MATH
  22. R. Sharma, “Second order parallel tensors on contact manifolds. II,” Comptes Rendus Mathématiques, vol. 13, no. 6, pp. 259–264, 1991. View at Zentralblatt MATH
  23. K. Yano and M. Ako, “Vector fields in Riemannian and Hermitian manifolds with boundary,” Kōdai Mathematical Seminar Reports, vol. 17, pp. 129–157, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. B. Ünal, “Divergence theorems in semi-Riemannian geometry,” Acta Applicandae Mathematicae, vol. 40, no. 2, pp. 173–178, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. G. Fubini, “Sulla teori degli spazii che ammettono un gruppo cinforme, Atti, Torino,” vol. 38, pp. 404–418, 1903.
  26. M. Obata, “The conjectures on conformal transformations of Riemannian manifolds,” Journal of Differential Geometry, vol. 6, pp. 247–258, 1971.
  27. J. Ferrand, “The action of conformal transformations on a Riemannian manifold,” Mathematische Annalen, vol. 304, no. 2, pp. 277–291, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. K. Yano and T. Nagano, “Einstein spaces admitting a one-parameter group of conformal transformations,” Annals of Mathematics, vol. 69, pp. 451–461, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. K. Yano, “Riemannian manifolds admitting a conformal transformation group,” Proceedings of the National Academy of Sciences of the United States of America, vol. 62, pp. 314–319, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. A. Lichnerowicz, “Sur les transformations conformes d'une variété riemannienne compacte,” Comptes Rendus de l'Académie des Sciences, vol. 259, pp. 697–700, 1964. View at Zentralblatt MATH
  31. A. Romero and M. Sánchez, “Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field,” Proceedings of the American Mathematical Society, vol. 123, no. 9, pp. 2831–2833, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. M. Romero, “The introduction of Bochner's technique on Lorentzian manifolds,” Nonlinear Analysis, vol. 47, no. 5, pp. 3047–3059, 2001. View at Publisher · View at Google Scholar
  33. W. Kühnel and H.-B. Rademacher, “Essential conformal fields in pseudo-Riemannian geometry,” Journal de Mathématiques Pures et Appliquées, vol. 74, no. 5, pp. 453–481, 1995. View at Zentralblatt MATH
  34. W. Kühnel and H.-B. Rademacher, “Conformal vector fields on pseudo-Riemannian spaces,” Differential Geometry and Its Applications, vol. 7, no. 3, pp. 237–250, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. D. Kramer, H. Stephani, M. MacCallum, and E. Herlt, Exact Solutions of Einstein's Field Equations, Cambridge University Press, Cambridge, UK, 1980.
  36. D. Eardley, J. Isenberg, J. Marsden, and V. Moncrief, “Homothetic and conformal symmetries of solutions of Einstein's equations,” Communications in Mathematical Physics, vol. 106, no. 1, pp. 137–158, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. R. Geroch, “Domain of dependence,” Journal of Mathematical Physics, vol. 11, pp. 437–449, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. C. N. Yang and R. D. Mills, “Isotropic spin and isotropic gauge invariance,” Physical Review, vol. 96, pp. 191–195, 1954.
  39. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, UK, 1973.
  40. E. M. Patterson, “On symmetric recurrent tensors of the second order,” The Quarterly Journal of Mathematics, vol. 2, pp. 151–158, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  41. G. S. Hall and J. da Costa, “Affine collineations in space-time,” Journal of Mathematical Physics, vol. 29, no. 11, pp. 2465–2472, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. G. S. Hall, A. D. Hossack, and J. R. Pulham, “Sectional curvature, symmetries, and conformally flat plane waves,” Journal of Mathematical Physics, vol. 33, no. 4, pp. 1408–1414, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of general relativity,” in Gravitation: An Introduction to Current Research, L. Witten, Ed., pp. 227–265, John Wiley & Sons, New York, NY, USA, 1962.
  44. R. Maartens and S. D. Maharaj, “Conformal killing vectors in Robertson-Walker spacetimes,” Classical and Quantum Gravity, vol. 3, no. 5, pp. 1005–1011, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. B. K. Berger, “Homothetic and conformal motions in spacelike slices of solutions of Einstein's equations,” Journal of Mathematical Physics, vol. 17, no. 7, pp. 1268–1273, 1976. View at Publisher · View at Google Scholar
  46. R. Sharma, “Conformal symmetries of Einstein's field equations and inital data,” Journal of Mathematical Physics, vol. 46, no. 4, pp. 1–8, 2005. View at Publisher · View at Google Scholar
  47. L. J. Alías, A. Romero, and M. Sánchez, “Spacelike hypersurfaces of constant mean curvature in certain spacetimes,” Nonlinear Analysis, vol. 30, no. 1, pp. 655–661, 1997. View at Publisher · View at Google Scholar
  48. K. L. Duggal and R. Sharma, “Conformal killing vector fields on spacetime solutions of Einstein's equations and initial data,” Nonlinear Analysis, vol. 63, no. 5 –7, pp. 447–454, 2005. View at Publisher · View at Google Scholar
  49. K. L. Duggal, “Affine conformal vector fields in semi-Riemannian manifolds,” Acta Applicandae Mathematicae, vol. 23, no. 3, pp. 275–294, 1991. View at Zentralblatt MATH
  50. Y. Tashiro, “On conformal collineations,” Mathematical Journal of Okayama University, vol. 10, pp. 75–85, 1960.
  51. D. P. Mason and R. Maartens, “Kinematics and dynamics of conformal collineations in relativity,” Journal of Mathematical Physics, vol. 28, no. 9, pp. 2182–2186, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  52. J. Marsden, “On completeness of homogeneous pseudo-riemannian manifolds,” Indiana University Mathematics, vol. 22, pp. 1065–1066, 1973.
  53. M. Gutiérrez, F. J. Palomo, and A. Romero, “A Berger-Green type inequality for compact Lorentzian manifolds,” Transactions of the American Mathematical Society, vol. 354, no. 11, pp. 4505–4523, 2002. View at Publisher · View at Google Scholar
  54. M. Gutiérrez, F. J. Palomo, and A. Romero, “Conjugate points along null geodesics on Lorentzian manifolds with symmetry,” in Proceedings of the Workshop on Geometry and Physics, pp. 169–182, Madrid, Spain, 2001.
  55. M. Gutiérrez, F. J. Palomo, and A. Romero, “Lorentzian manifolds with no null conjugate points,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 137, no. 2, pp. 363–375, 2004. View at Publisher · View at Google Scholar
  56. E. Hopf, “Closed surfaces without conjugate points,” Proceedings of the National Academy of Sciences of the United States of America, vol. 34, pp. 47–51, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  57. F. J. Palomo and A. Romero, “Conformally stationary Lorentzian tori with no conjugate points are flat,” Proceedings of the American Mathematical Society, vol. 137, no. 7, pp. 2403–2406, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  58. F. J. Palomo and A. Romero, “Compact conformally stationary Lorentzian manifolds with no causal conjugate points,” Annals of Global Analysis and Geometry, vol. 38, no. 2, pp. 135–144, 2010. View at Publisher · View at Google Scholar
  59. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364, Kluwer Academic, Dodrecht, The Netherlands, 1996.
  60. K. L. Duggal and D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific Publishing, River Edge, NJ, USA, 2007.
  61. K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Frontiers in Mathematics, Birkhäuser, Basel, Switzerland, 2010.
  62. B. Carter, “Killing horizons and orthogonally transitive groups in space-time,” Journal of Mathematical Physics, vol. 10, pp. 70–81, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  63. G. J. Galloway, “Maximum principles for null hypersurfaces and null splitting theorems,” Annales Henri Poincaré, vol. 1, no. 3, pp. 543–567, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  64. B. Carter, “Extended tensorial curvature analysis for embeddings and foliations,” in Geometry and Nature, vol. 203 of Contemporary Mathematics, pp. 207–219, American Mathematical Society, Providence, RI, USA, 1997. View at Zentralblatt MATH
  65. V. Perlick, “On totally umbilical submanifolds of semi-Riemannian manifolds,” Nonlinear Analysis, vol. 63, pp. 511–518, 2005.
  66. E. Gourgoulhon and J. L. Jaramillo, “A (1 + 3)-perspective on null hypersurfaces and isolated horizons,” Physics Reports, vol. 423, no. 4-5, pp. 159–294, 2006. View at Publisher · View at Google Scholar
  67. K. L. Duggal and R. Sharma, “Conformal evolution of spacetime solutions of Einstein's equations,” Communications in Applied Analysis, vol. 11, no. 1, pp. 15–22, 2007. View at Zentralblatt MATH
  68. S. Carlip, “Symmetries, horizons, and black hole entropy,” General Relativity and Gravitation, vol. 39, no. 10, pp. 1519–1523, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  69. T. Jacobson and G. Kang, “Conformal invariance of black hole temperature,” Classical and Quantum Gravity, vol. 10, no. 11, pp. L201–L206, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  70. J. Sultana and C. C. Dyer, “Conformal Killing horizons,” Journal of Mathematical Physics, vol. 45, no. 12, pp. 4764–4776, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  71. J. Sultana and C. C. Dyer, “Cosmological black holes: a black hole in the Einstein-de Sitter universe,” General Relativity and Gravitation, vol. 37, no. 8, pp. 1349–1370, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  72. K. L. Duggal, “Time-dependent black hole horizons on spacetime solutions of Einstein's equations with initial data,” in Advances in Lorentzian Geometry, M. Plaue and M. Scherfner, Eds., pp. 51–61, Aachen: Shaker, Berlin, Germany, 2008. View at Zentralblatt MATH