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On Almost Contingent Manifolds of Second Class with Applications in Relativity

Published online by Cambridge University Press:  20 November 2018

K. L. Duggal
K. L. Duggal*
Affiliation:
Department of Mathematics, University of Windsor, Windsor, Ontario N9B 3P4, Canada
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D. E. Blair [1] has introduced the notion of K-manifolds as an analogue of the even dimensional Kähler manifolds and of the odd dimensional quasi-Sasakian manifolds. These manifolds have been studied with respect to a positive definite metric. In this paper, we study a more general case of if-manifolds carrying an arbitrary non-degenerate metric, in particular, a metric of Lorentz signature. This theory is then applied within the frame-work of general relativity. Using the Ruse-Synge classification [8, 9] of non-null electromagnetic fields with source, we develop a geometric proof for the existence of either two space like or one space like and one time like Killing vector fields on the space-time manifold.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 3 , 01 September 1978 , pp. 289 - 295
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Blair, D. E., Geometry of manifolds with structural group U(n)XO(S), J. Difï. Geom. 4 (1970), 155-167.Google Scholar
2. Duggal, K. L., On a unified theory of differentiate structures, II: Existence theorems, Tensor, N.S. 29 (1975), 209-213.Google Scholar
3. Duggal, K. L., On the spheres carrying an almost contingent structure, Canad. Math. Bull. 18 (1975), 195-201.Google Scholar
4. Goldberg, S. I. and Yano, K., Polynomial structures on Manifolds, Kôdai Math. Sem. Rep., 22 (1970), 199-218.Google Scholar
5. Goldberg, S. I., Framed manifolds, differential geometry, in honour of K. Yano, Kinokuniya, Tokyo, (1972), 121-132.Google Scholar
6. Ishihara, S. and Yano, K., On the integrability conditions of a structure f, satisfying f 3 + f = 0, Quart. J. Math. Oxford 15 (1964) 217-222.Google Scholar
7. Michalski, H. and Wainwright, J., Killing vector fields and the Einstein-Maxwell field equations in general relativity, GRG Vol. 6 (1975), 289-318.Google Scholar
8. Ruse, H. S., On the geometry of the electromagnetic field in general relativity, Proc. London Math. Soc, 41 (1936), 302.Google Scholar
9. Synge, J. L., Principal null directions defined in space time by an electromagnetic field, U. of Toronto Studies in Applied Math. Sec, 1 (1935), 1-50.Google Scholar
10. Wooley, M. L., Structure of Groups of Motions admitted by Einstein-Maxwell Space-Times, Commun. Math. Phys., 31 (1973), 75-81.Google Scholar
11. Zund, J. D., Electromagnetic theory in general relativity III: The structure of sources, Tensor, N.S. 27 (1973), 355-360.Google Scholar