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Ricci curvature of submanifolds in Sasakian space forms

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  09 April 2009

Ion Mihai
Ion Mihai
Affiliation:
Faculty of Mathematics, University of Bucharest, Str. Academiei 14, 70109 Bucharest, Romania e-mail: imihai@math.math.unibuc.ro
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Abstract

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Recently, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Afterwards, we dealt with similar problems for submanifolds in complex space forms.

In the present paper, we obtain sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. Also, estimates of the scalar curvature and the k-Ricci curvature respectively, in terms of the squared mean curvature, are proved.

Keywords

Ricci curvaturemean curvatureSasakian space formC-totally real submanifoldcontact CR-submanifold.

MSC classification

Secondary: 53C40: Global submanifolds 53B25: Local submanifolds 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 2 , April 2002 , pp. 247 - 256
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509 (Springer, Berlin, 1976).CrossRefGoogle Scholar
[2]Chen, B. Y., ‘Some pinching and classification theorems for minimal submanifolds’. Arch. Math. 60 (1993), 568–578.CrossRefGoogle Scholar
[3]Chen, B. Y., ‘Mean curvature and shape operator of isometric immersions in real space form’, Glasgow Math. J. 38 (1996), 87–97.CrossRefGoogle Scholar
[4]Chen, B. Y., ‘Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions’, Glasgow Math. J. 41 (1999), 33–41.CrossRefGoogle Scholar
[5]Defever, F., Mihai, I. and Verstraelen, L., ‘B. Y. Chen's inequality for C-totally real submanifolds of Sasakian space forms’, Boll. Un. Mat. Ital. (7) 11 (1997), 365–374.Google Scholar
[6]Matsumoto, K., Mihai, I. and Oiaga, A., ‘Ricci curvature of submanifolds in complex space forms’, Rev. Roum. Math. Pures Appl. (to appear).Google Scholar
[7]Mihai, I., Rosca, R. and Verstraelen, L., Some aspects of the differential geometry of vector fields, Vol. 2 (PADGE, K. U. Leuven, K. U. Brussels, 1996).Google Scholar
[8]Yano, K. and Kon, M., Structures on manifolds (World Scientific, Singapore, 1984).Google Scholar