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On totally umbilical QR-submanifolds of quaternion Kaehlerian manifolds

Published online by Cambridge University Press:  17 April 2009

Aurel Bejancu  and
Hani Reda Farran
Aurel Bejancu
Affiliation:
Department of Mathematics and Computer Science, Kuwait Unviersity, PO Box 5969, Safat 13060, Kuwait e-mail: bejancu@math-1.sci.kuniv.edu.kwfarran@math-1.sci.kuniv.edu.kw
Hani Reda Farran
Affiliation:
Department of Mathematics and Computer Science, Kuwait Unviersity, PO Box 5969, Safat 13060, Kuwait e-mail: bejancu@math-1.sci.kuniv.edu.kwfarran@math-1.sci.kuniv.edu.kw
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Abstract

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We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 95 - 103
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Barros, M., Chen, B.Y. and Urbano, F., ‘Quaternion CR-submanifolds of quaternion manifolds’, Kodai Math. J. 4 (1981), 399–417.CrossRefGoogle Scholar
[2]Bejancu, A., Geometry of CR-submanifolds (Kluwer, Dordrecht, 1986).CrossRefGoogle Scholar
[3]Bejancu, A., ‘QR-submanifolds of quaternion Kaehlerian manifolds’, Chinese J. Math. 14 (1986), 81–94.Google Scholar
[4]Besse, A.L., Einstein manifolds (Springer-Verlag, Berlin, 1987).CrossRefGoogle Scholar
[5]Chen, B.Y., Geometry of submanifolds (Marcel Dekker, New York, 1973).Google Scholar
[6]Chen, B.Y., ‘Totally umbilical submanifolds of quaternion space forms’, J. Austral. Math. Soc. Ser. A 26 (1978), 154–162.CrossRefGoogle Scholar
[7]Ishihara, S., ‘Quaternion Kaehlerian manifolds’, J. Differential Geom. 9 (1974), 483–500.CrossRefGoogle Scholar
[8]Kuo, Y.Y., ‘On almost contact 3-structure’, Tôhoku Math. J. 22 (1970), 325–332.CrossRefGoogle Scholar
[9]Marchiafava, S., ‘Sulla geometria locale delle varietá Kähleriane quaternionali’, Boll. Un. Mat. Ital. 7 (1991), 417–447.Google Scholar
[10]Pak, J.S., ‘Anti-quaternionic submanifolds of a quaternion projective space’, Kyungpook Math. J. 18 (1981), 91–115.Google Scholar
[11]Udriste, C., ‘Structures presque coquaternionnes’, Bull. Math. Soc. Sci. Math. R.S. Roumanie 12 (1969), 487–507.Google Scholar