Conformally recurrent space-times admitting a proper conformal vector field
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper we study the properties of conformally recur- rent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field σ, 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 σㅎj 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 σj.
Abstract
In this paper we study the properties of conformally recur- rent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field , 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 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 j .
참고문헌 (34)
- T. Adati, "On Riemannian space with recurrent conformal curvature", Tensor (N.S.), 18, pp.348~354 (1967)
- A. Avez, "Formule de Gauss-Bonnet-Chern en metrique de signature quelconque", C. R. Acad. Sci. Paris, 255, pp.2049~2051 (1962)
- L. Bel, "Radiation states and the problem of energy in general relativity", Gen. Relativity Gravitation, 32(10), pp.2047~2078 (2000)
- B. Barua, "Proper conformal collineation in conformally recurrent spaces", Bull. Cal. Math. Soc, 91, pp.333~336 (1999)
- U. C. De, "On pseudo conformally symmetric manifolds", Bull. Calcutta Math. Soc, 85(5), pp.479~486 (1993)
- U. C. De, "Some remarks on proper conformal motions in pseudo conformally symmetric spaces", Tensor (N.S.), 60(1), pp.48~51 (1998)
- R. Debever, "Tenseur de super-energie, tenseur de Riemann.cas singuliers", C. R. Acad. Sci. (Paris), 249, pp.1744~1746 (1959)
- F. Defever, "On semi Riemannian manifolds satisfying the condition R · R = Q(S,R) in geometry and topology of submanifolds. III", pp.108~130 (1991), World Scientific Publ
- F. De Felice, "Relativity on Curved Manifolds" (1990), Cambridge University Press
- A. Derdzinski, "On compact manifolds admitting indefinite metrics with parallel Weyl tensor", J. Geom. Phys, 58(9), pp.1137~1147 (2008)
- A. Derdzinski, "Codazzi tensor fields, curvature and Pontryagin forms", Proc. London Math. Soc, 47(1), pp.15~26 (1983)
- G. S. Hall, "and Curvature Structure in General Relativity" (2004), World Scientific
- F. Hirzebruch, "New Topological Methods in Algebraic Topology" (1966), Springer
- V. R. Kaigorodov, "The curvature structure of spacetimes", Problems of Geometry, 14, pp.177~204 (1983)
- Quddus Khan, "On Recurrent Riemannian Manifolds", Kyungpook Mathematical Journal, 44(2), pp.269~276 (2004), 경북대학교 자연과학대학 수학과
- D. Lovelock, "Tensors, differential forms and variational principles" (1988), Dover
- C. A. Mantica, "A second order identity for the Riemannian tensor and applications", Colloq. Math, 122(1), pp.69~82 (2011)
- C. A. Mantica, "Extended Derdzinski-Shen theorem for curvature tensors", Colloq. Math, 128(1), pp.1~6 (2012)
- C. A. Mantica, "Riemann compatible tensors, Colloq. Math. 128 (2012), no. 2, 197–210", Colloq. Math, 128(2), pp.197~210 (2012)
- "2013", ~
- "2013", ~
- R. G. McLenaghan, "Complex recurrent space-times", Proc. Roy. Soc. London Ser. A, 327, pp.229~249 (1972)
- R. G. McLenaghan, "Second order recurrent space-times in general relativity", Lett. Nuovo Cimento, 5(7), pp.563~564 (1972)
- M. Nakahara, "Geometry, Topology and Physics" (2003), Taylor & Francis
- A. Z. Petrov, "The classification of spaces defining gravitational field", Gen. Relativity Gravitation, 32(8), pp.1665~1685 (2000)
- M. M. Postnikov, "Geometry VI: Riemannian geometry" (2001), Springer
- R. Sachs, "Gravitational waves in general relativity. VI. The outgoing radiation condition", Proc. Roy. Soc. Ser. A, 264, pp.309~338 (1961)
- R. Sharma, "Proper conformal symmetries of conformal symmetric spaces", J. Math. Phys, 29(11), pp.2421~2422 (1988)
- R. Sharma, "Proper conformal symmetries of space-times with divergence-free Weyl conformal tensor", J. Math. Phys, 34(8), pp.3582~3587 (1988)
- H. Sthepani, "General Relativity" (2004), Cambridge University Press
- H. Sthepani, "Exact solutions of Einstein’s Field Equations" (2003), Cambridge University Press
- Y. J. Suh, "Conformally recurrent semi-Riemannian manifolds", Rocky Mountain J. Math, 35(1), pp.285~307 (2005)
- A. G. Walker, "On Ruse’s spaces of recurrent curvature", Proc. London Math. Soc, 52, pp.36~64 (1950)
- K. Yano, "The Theory of Lie Derivatives and Its applications" (1957), Interscience