Journal Menu

- About this Journal ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

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

- 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 - B. O'Neill,
*Semi-Riemannian Geometry with Applications to Relativity*, Academic Press, New York, NY, USA, 1983. View at Zentralblatt MATH - 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 - D. N. Kupeli,
*Singular Semi-Riemannian Geometry*, vol. 366, Kluwer Academic, Dodrecht, The Netherlands, 1996. View at Zentralblatt MATH - 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 - 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 - 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 - 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 - 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 - 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 - K. Yano,
*Integral Formulas in Riemannian Geometry*, Marcel Dekker, New York, NY, USA, 1970. - 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 - 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 - 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 - K. Yano,
*Differential Geometry on Complex and Almost Complex Spaces*, Pergamon Press, New York, NY, USA, 1965. - 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 - K. L. Duggal and R. Sharma,
*Symmetries of Spacetimes and Riemannian Manifolds*, vol. 487, Kluwer Academic, Dodrecht, The Netherlands, 1999. - D. E. Blair,
*Contact Manifolds in Riemannian Geometry*, vol. 509 of*Lecture notes in Math*, Springer, Berlin, Germany, 1976. - 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 - 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 - 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 - 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 - 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 - 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 - G. Fubini, “Sulla teori degli spazii che ammettono un gruppo cinforme, Atti, Torino,” vol. 38, pp. 404–418, 1903.
- M. Obata, “The conjectures on conformal transformations of Riemannian manifolds,”
*Journal of Differential Geometry*, vol. 6, pp. 247–258, 1971. - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - D. Kramer, H. Stephani, M. MacCallum, and E. Herlt,
*Exact Solutions of Einstein's Field Equations*, Cambridge University Press, Cambridge, UK, 1980. - 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 - 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 - C. N. Yang and R. D. Mills, “Isotropic spin and isotropic gauge invariance,”
*Physical Review*, vol. 96, pp. 191–195, 1954. - S. W. Hawking and G. F. R. Ellis,
*The Large Scale Structure of Space-Time*, Cambridge University Press, Cambridge, UK, 1973. - 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 - 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 - 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 - 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. - 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 - 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 - 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 - 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 - 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 - 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 - Y. Tashiro, “On conformal collineations,”
*Mathematical Journal of Okayama University*, vol. 10, pp. 75–85, 1960. - 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 - J. Marsden, “On completeness of homogeneous pseudo-riemannian manifolds,”
*Indiana University Mathematics*, vol. 22, pp. 1065–1066, 1973. - 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 - 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. - 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 - 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 - 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 - 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 - K. L. Duggal and A. Bejancu,
*Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications*, vol. 364, Kluwer Academic, Dodrecht, The Netherlands, 1996. - K. L. Duggal and D. H. Jin,
*Null Curves and Hypersurfaces of Semi-Riemannian Manifolds*, World Scientific Publishing, River Edge, NJ, USA, 2007. - K. L. Duggal and B. Sahin,
*Differential Geometry of Lightlike Submanifolds*, Frontiers in Mathematics, Birkhäuser, Basel, Switzerland, 2010. - 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 - 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 - 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 - V. Perlick, “On totally umbilical submanifolds of semi-Riemannian manifolds,”
*Nonlinear Analysis*, vol. 63, pp. 511–518, 2005. - 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 - 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 - 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 - 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 - 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 - 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 - 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