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On a characterisation of conformally-flat Riemannian spaces of class one

Published online by Cambridge University Press:  09 April 2009

R. N. Sen
R. N. Sen
Affiliation:
Department of Pure Mathematics Calcutta University
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In a previous paper [1] we considered those conformally-flat Riemannian spaces which satisfy the tensorial characterisation where, as usual, gij, Rhijk, Rij are the fundamental tensor, the curvature tensor, the Ricci tensor and E ≠ 0, F are certain scalars. The tensor g is always supposed to be real and analytic. A special form of the metrics of these spaces was seen to be where f is any real analytic function, subject to a restriction, of the argument θ. Writing f, f′, f″,… for f(θ), df|dθ, d2f|dθ2, … the quantities E, F and the scalar curvature R of the type of spaces (1.2) were seen to be

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 2 , May 1966 , pp. 172 - 178
Copyright
Copyright © Australian Mathematical Society 1966

References

[1]Sen, R. N., ‘On a type of Riemannian space conformal to a flat space’, Journ. Ind. Math. Soc. 21 (1957), 104–114.Google Scholar
[2]Levine, J., ‘Fields of parallel vectors in conformally flat spaces’, Duke Math. Journ. 17 (1950), 15–20.CrossRefGoogle Scholar
[3]Wong, Y., ‘Family of totally umbilical hypersurfaces in an Einstein space’, Annals of Math. (2) 44 (1943), 271–297.CrossRefGoogle Scholar
[4]Verbickii, L. L., ‘Geometry of conformal Euclidean spaces of class one’, Transactions of the seminar of vector and tensor analysis (translated from Russian), IX (1952), 146–182.Google Scholar
[5]Matsumoto, M., ‘Conformally flat Riemannian spaces of class one’, Journ. Math. Soc. of Japan, 3 (1951), 306–309.Google Scholar