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Projective-symmetric spaces

Published online by Cambridge University Press:  09 April 2009

R. F. Reynolds  and
A. H. Thompson
R. F. Reynolds
Affiliation:
Department of Mathematics University of Pittsburgh Pittsburgh, Pennsylvania
A. H. Thompson
Affiliation:
Department of Mathematics University of Pittsburgh Pittsburgh, Pennsylvania
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Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 1 , February 1967 , pp. 48 - 54
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Soos, Gy., ‘Ueber die geodaetischen Abbildung von Riemannsche Raeumen auf projectivsymmetrische Riemannsche Raeume’, Acta Math. Acad. Sci. Hung., 9 (1958), 349–361.Google Scholar
[2]Gupta, B., ‘On Projective-Symmetric Spaces’, J. Austr. Math. Soc., 4 (1964), 113–121.Google Scholar
[3]Schouten, J. A., Ricci calculus, 2nd Ed., Springer-Verlag, Berlin (1954), 163.Google Scholar
[4]Eisenhart, L. P., Riemannian geometry, Princeton U. Press, Princeton (1926).Google Scholar
[5]Eisenhart, L. P., Non-Riemannian geometry, Amer. Math. Soc., New York (1927).Google Scholar
[6]Sinjukow, N. S., ‘On the geodesic correspondence of a Riemannian space with a symmetric space’ (Russian) D. A. N. (U.S.S.R.), 98 (1954), 21–23.Google Scholar
[7]Weyl, H., ‘Reine infinitesimal Geometrie’, Math. Zeitschrift, 2 (1918), 384–411.CrossRefGoogle Scholar