Spacetimes admitting quasi-conformal curvature tensor

‎Mallick‎, S.; Zhao, P.; De, U. C.

Spacetimes admitting quasi-conformal curvature tensor

Document Type : Research Paper

Authors

¹ Department of Mathematics‎, ‎Chakdaha College‎, ‎P‎. ‎O‎. ‎Chakdaha‎, ‎Dist-Nadia‎, ‎West Bengal‎, ‎India.

² Department of Applied Mathematics‎, ‎Nanjing University of Science and Technology‎, ‎Nanjing 210094‎, ‎P‎. ‎R‎. ‎China.

³ Department of Pure Mathematics‎, ‎University of Calcutta‎, ‎35‎, ‎B‎. ‎C‎. ‎Road‎, ‎Kolkata 700019‎, ‎West Bengal‎, ‎India.

Abstract

‎The object of the present paper is to study spacetimes admitting‎ ‎quasi-conformal curvature tensor‎. ‎At first we prove that a quasi-conformally flat spacetime is Einstein‎ ‎and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying‎ ‎Einstein's field equation with cosmological constant is covariant constant‎. ‎Next‎, ‎we prove that if the perfect fluid pacetime with‎ ‎vanishing quasi-conformal curvature tensor obeys Einstein's field equation without cosmological constant‎, ‎then the spacetime has constant energy density and isotropic pressure and the perfect fluid always behave‎ ‎as a cosmological constant and also such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field $U$‎. ‎Moreover‎, ‎it is shown that in a purely electromagnetic distribution the spacetime with vanishing quasi-conformal curvature tensor is filled with‎ ‎radiation and extremely hot gases‎. ‎We also study dust-like fluid spacetime with vanishing quasi-conformal curvature tensor.

Keywords

Main Subjects

53-XX Differential geometry

References

Z. Ahsan and S. A. Siddiqui, Concircular Curvature Tensor and Fluid Spacetimes, Int. J. Theor. Phys. 48 (2009), no. 11, 3202--3212.

L. Amendola and S. Tsujikawa, Dark Energy: Theory and Observations, Cambridge Univ. Press, Cambridge, 2010.

K. Amur and Y. B. Maralabhavi, On quasi-conformally at spaces, Tensor (N.S.) 31 (1977), no. 2, 194--198.

M. C. Chaki and S. Roy, Space-times with covariant-constant energy-momentum tensor, Internat. J. Theoret. Phys. 35 (1996), no. 5, 1027--1032.

S. Chakraborty, N. Mazumder and R. Biswas, Cosmological evolution across phantom crossing and the nature of the horizon, Astrophys. Space Sci. Libr. 334 (2011) 183--186.

U. C. De and A. Sarkar, On the quasi-conformal curvature tensor of a (k; _)-contact metric manifold, Math. Rep. 14 (2012), no. 2, 115--129.

U. C. De and Y. Matsuyama, Quasi-conformally at manifolds satisfying certain conditions on the Ricci tensor, SUT J. Math. 42 (2006), no. 2, 295--303.

U. C. De, J. B. Jun and A. K. Gazi, Sasakian manifolds with quasi-conformal curvature tensor, Bull. Korean Math. Soc. 45 (2008), no. 2, 313--319.

U. C. De and L. Velimirovic, Spacetimes with semisymmetric energy momentum tensor, Int. J. Theor. Phys. 54 (2015), no. 6, 1779--1783.

K. L. Duggal, Curvature Collineations and Conservation Laws of General Relativity, presented at the Canadian conference on general relativity and relativistic Astro-physics, Halifax, Canada, 1985.

K. L. Duggal, Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times, J. Math. Phys. 33 (1992), no. 9, 2989--2997.

S. Guha, On a perfect uid space-time admitting quasi conformal curvature tensor, Facta Univ. Ser. Mech. Automat. Control Robot. 3 (2003), no. 14, 843--849.

A. Hosseinzadeh and A. Taleshian, On conformal and quasi-conformal curvature tensors of an N(k)-quasi Einstein manifold, Commun. Korean Math. Soc. 27 (2012), no. 2, 317--326.

H. Karchar, Infinitesimal characterization of Friedmann universe, Arch. Math. (Basel) 38 (1992), no. 1, 58--64.

G. H. Katzin, J. Levine and W. R. Davis, Curvature collineations: A fundamental symmetry property of the spacetime of general relativity defined by the vanishing Lie derivative of the Riemannian curvature tensor, J. Math. Phys. 10 (1969) 617--629.

C. A. Mantica and Y. J. Suh, Conformally symmetric manifolds and quasi conformally recurrent Riemannian manifolds, Balkan J. Geom. Appl. 16 (2011), no. 1, 66--77.

C. A. Mantica and Y. J. Suh, Pseudo-Z symmetric space-times, J. Math. Phys. 55 (2014), no. 4, 12 pages.

J. V. Narlikar, General Relativity and Gravitation, The Macmillan Co. India, 1978.

B. O'Neill, Semi-Riemannian Geometry, Academic Press, NY, 1983.

R. K. Sach and W. Hu, General Relativity for Mathematician, Springer Verlag, New York, 1977.

S. K. Srivastava, General Ralativity and Cosmology, Prentice-Hall of India, Private Limited, New Delhi, 2008.

C. Ozgur and S. Sular, On N(k)-quasi-Einstein manifolds satisfying certain conditions, Balkan J. Geom. Appl. 13 (2008) 74--79.

H. Stephani, General Relativity-An Introduction to the Theory of Gravitational Field, Cambridge Univ. Press, Cambridge, 1982.

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact Solutions of Einstein's Field Equations, Cambridge Monogr. on Math. Phys. Cambridge Univ. Press, 2nd edition, Cambridge, 2003.

K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transformation group, J. Differential Geom. 2 (1968) 161--184.

F. O. Zengin, M-projectively at spacetimes, Math. Rep. 4 (2012), no. 4, 363--370.