Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CONFORMALLY RECURRENT SPACE-TIMES ADMITTING A PROPER CONFORMAL VECTOR FIELD

  • Received : 2013.11.03
  • Published : 2014.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we study the properties of conformally recurrent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field ${\sigma}$, focusing particularly on the 4-dimensional Lorentzian case. Some general properties already proven by one of the present authors for pseudo conformally symmetric manifolds endowed with a conformal vector field are proven also in the case, and some new others are stated. Moreover interesting results are pointed out; for example, it is proven that the Ricci tensor under certain conditions is Weyl compatible: this notion was recently introduced and investigated by one of the present authors. Further we study conformally recurrent 4-dimensional Lorentzian manifolds (space-times) admitting a conformal vector field: it is proven that the covector ${\sigma}_j$ is null and unique up to scaling; moreover it is shown that the same vector is an eigenvector of the Ricci tensor. Finally, it is stated that such space-time is of Petrov type N with respect to ${\sigma}_j$.

Keywords

References

  1. T. Adati and T. Miyazawa, On Riemannian space with recurrent conformal curvature, Tensor (N.S.) 18 (1967), 348-354.
  2. A. Avez, Formule de Gauss-Bonnet-Chern en metrique de signature quelconque, C. R. Acad. Sci. Paris 255 (1962), 2049-2051.
  3. L. Bel, Radiation states and the problem of energy in general relativity, Gen. Relativity Gravitation 32 (2000), no. 10, 2047-2078. https://doi.org/10.1023/A:1001958805232
  4. B. Barua and U. C. De, Proper conformal collineation in conformally recurrent spaces, Bull. Cal. Math. Soc. 91 (1999), 333-336.
  5. U. C. De and H. A. Biswas, On pseudo conformally symmetric manifolds, Bull. Calcutta Math. Soc. 85 (1993), no. 5, 479-486.
  6. U. C. De and B. K. Mazumder, Some remarks on proper conformal motions in pseudo conformally symmetric spaces, Tensor (N.S.) 60 (1998), no. 1, 48-51.
  7. R. Debever, Tenseur de super-energie, tenseur de Riemann.cas singuliers, C. R. Acad. Sci. (Paris) 249 (1959), 1744-1746.
  8. F. Defever and R. Deszcz, On semi Riemannian manifolds satisfying the condition R ${\cdot}$ R = Q(S,R) in geometry and topology of submanifolds. III, World Scientific Publ. Singapore (1991), 108-130.
  9. F. De Felice and C. J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, 1990.
  10. A. Derdzinski and W. Roter, On compact manifolds admitting indefinite metrics with parallel Weyl tensor, J. Geom. Phys. 58 (2008), no. 9, 1137-1147. https://doi.org/10.1016/j.geomphys.2008.03.011
  11. A. Derdzinski and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc. 47 (1983), no. 1, 15-26.
  12. G. S. Hall, Symmetries and Curvature Structure in General Relativity, World Scientific Singapore, 2004.
  13. F. Hirzebruch, New Topological Methods in Algebraic Topology, Springer, 1966.
  14. V. R. Kaigorodov, The curvature structure of spacetimes, Problems of Geometry 14 (1983), 177-204.
  15. Q. Khan, On Recurrent Riemannian Manifolds, Kyungpook Math. J. 44 (2004), no. 2, 269-276.
  16. D. Lovelock and H. Rund, Tensors, differential forms and variational principles, reprint Dover ed 1988.
  17. C. A. Mantica and L. G. Molinari, A second order identity for the Riemannian tensor and applications, Colloq. Math. 122 (2011), no. 1, 69-82. https://doi.org/10.4064/cm122-1-7
  18. C. A. Mantica and L. G. Molinari, Extended Derdzinski-Shen theorem for curvature tensors, Colloq. Math. 128 (2012), no. 1, 1-6. https://doi.org/10.4064/cm128-1-1
  19. C. A. Mantica and L. G. Molinari, Riemann compatible tensors, Colloq. Math. 128 (2012), no. 2, 197-210. https://doi.org/10.4064/cm128-2-5
  20. C. A. Mantica and L. G. Molinari, Weyl compatible tensors, arXiv: 1212.1273 [math-ph], 21 Jan. 2013.
  21. C. A. Mantica and L. G. Molinari, Conformally quasi recurrent pseudo-Riemannian manifolds, arXiv: 1305.5060 vl [math. D. G.], 22 May 2013.
  22. R. G. McLenaghan and J. Leroy, Complex recurrent space-times, Proc. Roy. Soc. London Ser. A 327 (1972), 229-249. https://doi.org/10.1098/rspa.1972.0042
  23. R. G. McLenaghan and H. A. Thompson, Second order recurrent space-times in general relativity, Lett. Nuovo Cimento 5 (1972), no. 7, 563-564. https://doi.org/10.1007/BF02752675
  24. M. Nakahara, Geometry, Topology and Physics, Second Edition, Taylor & Francis, New York, 2003.
  25. A. Z. Petrov, The classification of spaces defining gravitational field, Gen. Relativity Gravitation 32 (2000), no. 8, 1665-1685. https://doi.org/10.1023/A:1001910908054
  26. M. M. Postnikov, Geometry VI: Riemannian geometry, Encyclopaedia of Mathematical Sciences, Vol. 91, Springer, 2001.
  27. R. Sachs, Gravitational waves in general relativity. VI. The outgoing radiation condition, Proc. Roy. Soc. Ser. A 264 (1961), 309-338. https://doi.org/10.1098/rspa.1961.0202
  28. R. Sharma, Proper conformal symmetries of conformal symmetric spaces, J. Math. Phys. 29 (1988), no. 11, 2421-2422. https://doi.org/10.1063/1.528127
  29. R. Sharma, Proper conformal symmetries of space-times with divergence-free Weyl conformal tensor, J. Math. Phys. 34 (1988), no. 8, 3582-3587.
  30. H. Sthepani, General Relativity, Cambridge University Press, 2004.
  31. H. Sthepani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Hertl, Exact solutions of Einstein's Field Equations, Cambridge University Press, 2003.
  32. Y. J. Suh and J.-H. Kwon, Conformally recurrent semi-Riemannian manifolds, Rocky Mountain J. Math. 35 (2005), no. 1, 285-307. https://doi.org/10.1216/rmjm/1181069782
  33. A. G. Walker, On Ruse's spaces of recurrent curvature, Proc. London Math. Soc. 52 (1950), 36-64.
  34. K. Yano, The Theory of Lie Derivatives and Its applications, Interscience, New York, 1957.