Abstract
Warped product C R-submanifolds in Kählerian manifolds were intensively studied only since 2001 after the impulse given by B. Y. Chen in [3], [4]. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kählerian geometry, was developed, namely warped product contact C R-submanifolds in Sasakian manifolds (cf. [7], [8]). In this note we prove that there exists no proper doubly warped product contact C R-submanifolds in trans-Sasakian manifolds.
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References
K. Arslan, R. Ezentas, I. Mihai and C. Murathan, Contact warped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc., 42 (2005), 1101–1110.
D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, vol. 203, Birkhäuser (Boston, Basel, Berlin, 2001).
B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math., 133 (2001), 177–195.
B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II, Monatsh. Math., 134 (2001), 103–119.
A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123 (1980), 35–58.
J. C. Marero, The local structure of trans-Sasakian manifolds, Ann. Mat. Pura Appl., 162 (1992), 77–86.
I. Mihai, Contact CR-warped product submanifolds in Sasakian space forms, Geom. Dedicata, 109 (2004), 165–173.
M. I. Munteanu, Warped product contact CR-submanifolds of Sasakian space forms, Publ. Math. Debrecen, 66 (2005), 75–120.
B. Ünal, Doubly warped products, Differential Geometry and Its Applications, 15 (2001), 253–263.
K. Yano and M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds, Birkhäuser (Boston, Basel, Stuttgart, 1983).
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The author was a beneficiary of a Research Fellowship of the Ministerie van de Vlaamse Gemeenschap, ref. 13DA-IZ/JDW/ROE/2004-2005/04-2439 at the Katholieke Universiteit Leuven, Belgium.
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Munteanu, M.I. A note on doubly warped product contact C R-submanifolds in trans-Sasakian manifolds. Acta Math Hung 116, 121–126 (2007). https://doi.org/10.1007/s10474-007-6013-x
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DOI: https://doi.org/10.1007/s10474-007-6013-x
Key words and phrases
- trans-Sasakian manifolds
- Sasakian manifolds
- Kenmotsu manifolds
- doubly warped product
- contact CR-submanifolds