## Concircular curvature tensor and fluid spacetimes.(English)Zbl 1255.83110

Summary: In the differential geometry of certain $$F$$-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.

### MSC:

 83E05 Geometrodynamics and the holographic principle 83C15 Exact solutions to problems in general relativity and gravitational theory 53Z05 Applications of differential geometry to physics
Full Text:

### References:

 [1] Ahsan, Z.: On a geometrical symmetry of the spacetime of general relativity. Bull. Cal. Math. Soc. 97(3), 191–200 (2005) · Zbl 1089.83005 [2] Blair, D.E., Kim, J.-S., Tripathi, M.M.: On the concircular curvature tensor of a contact metric manifold. J. Korean Math. Soc. 42(5), 883–892 (2005) · Zbl 1084.53039 [3] Derdzinski, A., Shen, C.-L.: Codazzi tensor fields, curvature and Pontryagin forms. Proc. Lond. Math. Soc. 47(3), 15–26 (1983) · Zbl 0519.53015 [4] Ellis, G.F.R.: Relativistic cosmology. In: Sachs, R.K. (ed.) General Relativity and Cosmology. Academy Press, New York (1971) · Zbl 0337.53058 [5] Kalligas, D., Wesson, P., Everitt, C.W.F.: Flat FRW models with variable G and {$$\Lambda$$}. Gen. Relativ. Gravit. 24, 351 (1992) [6] Narlikar, J.V.: General Relativity and Gravitation. The Macmillan Co. of India (1978) [7] Stephani, H.: General Relativity–An Introduction to the Theory of Gravitational Field. Cambridge University Press, Cambridge (1982) · Zbl 0494.53026 [8] Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251 (1965) · Zbl 0136.17701 [9] Yano, K.: Concircular geometry I. Concircular transformations. Proc. Imp. Acad. Tokyo 16, 195–200 (1940) · Zbl 0024.08102 [10] Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) · Zbl 0213.23801 [11] Yano, K., Kon, M.: Structures on Manifolds. World Scientific Publishing, Singapore (1984) · Zbl 0557.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.