×

On submersion of CR-submanifolds of l.c.q.K. manifold. (English) Zbl 1256.32035

Summary: We study submersion of CR-submanifolds of an l.c.q.K. manifold. We show that if an almost Hermitian manifold \(B\) admits a Riemannian submersion \(\pi : M \to B\) of a CR-submanifold \(M\) of a locally conformal quaternion Kähler manifold \(\overline{M}\), then \(B\) is a locally conformal quaternion Kähler manifold.

MSC:

32V99 CR manifolds
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
Full Text: DOI

References:

[1] I. Vaisman, “On locally conformal almost kähler manifolds,” Israel Journal of Mathematics, vol. 24, no. 3-4, pp. 338-351, 1976. · Zbl 0335.53055 · doi:10.1007/BF02834764
[2] S. Dragomir and L. Ornea, Locally Conformal Kähler Geometry, vol. 155 of Progress in Mathematics, Birkhäauser, Boston, Mass, USA, 1998. · Zbl 0887.53001
[3] L. Ornea and P. Piccinni, “Locally conformal kähler structures in quaternionic geometry,” Transactions of the American Mathematical Society, vol. 349, no. 2, pp. 641-655, 1997. · Zbl 0865.53038 · doi:10.1090/S0002-9947-97-01591-2
[4] L. Ornea, “Weyl structure on quaternioric manifolds, a state of the art,” http://arxiv.org/abs/math/0105041.
[5] B. Sahin and R. Günes, “QR-submanifolds of a locally conformal quaternion kaehler manifold,” Publicationes Mathematicae Debrecen, vol. 63, no. 1-2, pp. 157-174, 2003. · Zbl 1027.53060
[6] B. O’Neill, “The fundamental equations of a submersion,” The Michigan Mathematical Journal, vol. 13, pp. 459-469, 1966. · Zbl 0145.18602 · doi:10.1307/mmj/1028999604
[7] A. Bejancu, “ CR submanifolds of a kaehler manifold. I,” Proceedings of the American Mathematical Society, vol. 69, no. 1, pp. 135-142, 1978. · Zbl 0368.53040 · doi:10.2307/2043207
[8] S. Kobayashi, “Submersions of CR submanifolds,” The Tohoku Mathematical Journal, vol. 39, no. 1, pp. 95-100, 1987. · Zbl 0619.58004 · doi:10.2748/tmj/1178228282
[9] S. Deshmukh, T. Ghazal, and H. Hashem, “Submersions of CR-submanifolds on an almost hermitian manifold,” Yokohama Mathematical Journal, vol. 40, no. 1, pp. 45-57, 1992. · Zbl 0772.53043
[10] J. Simons, “Minimal varieties in riemannian manifolds,” Annals of Mathematics, vol. 88, no. 1, pp. 62-105, 1968. · Zbl 0181.49702 · doi:10.2307/1970556
[11] R. Al-Ghefari, M. H. Shahid, and F. R. Al-Solamy, “Submersion of CR-submanifolds of locally conformal kaehler manifold,” Contributions to Algebra and Geometry, vol. 47, no. 1, pp. 147-159, 2006. · Zbl 1097.53049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.