Generalized Sasakian-space-forms. (English) Zbl 1064.53026

The authors introduce generalized Sasakian-space-forms and study their basic properties. They state that every generalized Sasakian-space-form with a \(K\)-contact structure is a Sasakian manifold, and, if the dimension is \(\geq5\), a Sasakian-space-form. The conditions for a generalized Sasakian-space-form to be a contact metric manifold are investigated. To construct many examples of these manifolds the authors use a wide variety of geometric constructions, such as Riemannian submersions, product manifolds, warped products, conformal transformations, \(D\)-homothetic deformations and \(D\)-conformal deformations. Some further results on generalized complex-space-forms are also stated.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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[1] Blair, D. E., The theory of quasi-Sasakian structures, Journal of Differential Geometry, 1, 331-345 (1967) · Zbl 0163.43903
[2] Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds (2002), Boston: Birkhäuser, Boston · Zbl 1011.53001
[3] Bueken, P.; Vanhecke, L., Curvature characterizations in contact geometry, Rivista di Matematica della Università di Parma, 14, 4, 303-313 (1988) · Zbl 0689.53021
[4] Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type (1984), Singapore: World Scientific, Singapore · Zbl 0537.53049
[5] S. Ianus and D. Smaranda,Some remarkable structures on the product of an almost contact metric manifold with the real line, inNational Colloquium on Geometry and Topology, University of Timişoara, 1977, pp. 107-110.
[6] Janssens, D.; Vanhecke, L., Almost contact structures and curvature tensors, Kodai Mathematical Journal, 4, 1-27 (1981) · Zbl 0472.53043
[7] Kenmotsu, K., A class of almost contact Riemannian manifolds, Tôhoku Mathematical Journal, 24, 93-103 (1972) · Zbl 0245.53040
[8] Ludden, G. D., Submanifolds of cosymplectic manifolds, Journal of Differential Geometry, 4, 237-244 (1970) · Zbl 0197.47902
[9] Marrero, J. C., The local structure of trans-Sasakian manifolds, Annali di Matematica Pura ed Applicata, 162, 77-86 (1992) · Zbl 0772.53036
[10] Olszak, Z., On the existence of generalized complex space forms, Israel Journal of Mathematics, 65, 214-218 (1989) · Zbl 0674.53061
[11] O’Neill, B., Semi-Rimannian Geometry with Applications to Relativity (1983), New York: Academic Press, New York
[12] Oubiña, J. A., New classes of almost contact metric structures, Publications Mathematicae Debrecen, 32, 187-193 (1985) · Zbl 0611.53032
[13] Sharma, R., On the curvature of contact metric manifolds, Journal of Geometry, 53, 179-190 (1995) · Zbl 0833.53033
[14] Suguri, S.; Nakayama, S., D-conformal deformations on almost contact metric structure, Tensor. New Series, 28, 125-129 (1974) · Zbl 0288.53034
[15] Tanno, S., The topology of contact Riemannian manifolds, Illinois Journal of Mathematics, 12, 700-717 (1968) · Zbl 0165.24703
[16] Tricerri, F.; Vanhecke, L., Curvature tensors on almost Hermitian manifolds, Transactions of the American Mathematical Society, 267, 365-398 (1981) · Zbl 0484.53014
[17] Vaisman, I., Conformal changes of almost contact metric structures, Geometry and Differential Geometry, 435-443 (1980), Berlin: Springer-Verlag, Berlin
[18] Vanhecke, L., Almost Hermitian manifolds with J-invariant Riemann curvature tensor, Rendiconti del Seminario Mathematico della Universitá e Politecnico di Torino, 34, 487-498 (1975)
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