We consider a Riemannian submersion π: M → N, where M is a CR-submanifold of a locally conformal Kaehler manifold L with the Lee form ω which is strongly non-Kaehler and N is an almost Hermitian manifold. First, we study some geometric structures of N and the relation between the holomorphic sectional curvatures of L and N. Next, we consider the leaves M of the foliation given by ω = 0 and give a necessary and sufficient condition for M to be a Sasakian manifold.
@article{bwmeta1.element.bwnjournal-article-cmv70i2p165bwm,
author = {Fumio Narita},
title = {CR-submanifolds of locally conformal Kaehler manifolds and Riemannian submersions},
journal = {Colloquium Mathematicae},
volume = {70},
year = {1996},
pages = {165-179},
zbl = {0860.53035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv70i2p165bwm}
}
Narita, Fumio. CR-submanifolds of locally conformal Kaehler manifolds and Riemannian submersions. Colloquium Mathematicae, Tome 70 (1996) pp. 165-179. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv70i2p165bwm/
[000] [1] R. L. Bishop and S. I. Goldberg, On the topology of positively curved Kaehler manifolds II, Tôhoku Math. J. 17 (1965), 310-318. | Zbl 0134.17906
[001] [2] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer, Berlin, 1976. | Zbl 0319.53026
[002] [3] D. E. Blair and B. Y. Chen, On CR-submanifolds of Hermitian manifolds, Israel J. Math. 34 (1979), 353-363. | Zbl 0453.53018
[003] [4] B. Y. Chen and L. Vanhecke, Isometric, holomorphic and symplectic reflections, Geom. Dedicata 29 (1989), 259-277. | Zbl 0673.53035
[004] [5] S. Dragomir, On submanifolds of Hopf manifolds, Israel J. Math. (2) 61 (1988), 98-110.
[005] [6] S. Dragomir, Cauchy-Riemann submanifolds of locally conformal Kaehler manifolds, I-II, Geom. Dedicata 28 (1988), 181-197, Atti Sem. Mat. Fis. Univ. Modena 37 (1989), 1-11. | Zbl 0659.53041
[006] [7] S. Kobayashi, Submersions of CR submanifolds, Tôhoku Math. J. 89 (1987), 95-100. | Zbl 0619.58004
[007] [8] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1, 2, Interscience Publishers, 1963, 1969. | Zbl 0119.37502
[008] [9] F. Narita, Riemannian submersions and isometric reflections with respect to submanifolds, Math. J. Toyama Univ. 15 (1992), 83-94. | Zbl 0776.53026
[009] [10] F. Narita, Riemannian submersion with isometric reflections with respect to the fibers, Kodai Math. J. 16 (1993), 416-427. | Zbl 0795.53042
[010] [11] B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 1-20.
[011] [12] R. C. Randell, Generalized Brieskorn manifolds, Bull. Amer. Math. Soc. 80 (1974), 111-115. | Zbl 0297.57018
[012] [13] T. Takahashi, Deformations of Sasakian structures and its application to the Brieskorn manifolds, Tôhoku Math. J. 30 (1978), 37-43. | Zbl 0392.53025
[013] [14] Y. Tashiro, On contact structure of hypersurfaces in complex manifolds, I, ibid. 15 (1963), 62-78. | Zbl 0113.37204
[014] [15] I. Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math. 24 (1976), 338-351. | Zbl 0335.53055
[015] [16] I. Vaisman, A theorem on compact locally conformal Kähler manifolds, Proc. Amer. Math. Soc. 75 (1979), 279-283. | Zbl 0414.53045
[016] [17] I. Vaisman, Locally conformal Kähler manifolds with parallel Lee form, Rend. Mat. 12 (1979), 263-284. | Zbl 0447.53032
[017] [18] I. Vaisman, Some curvature properties of locally conformal Kaehler manifolds, Trans. Amer. Math. Soc. (2) 259 (1980), 439-447. | Zbl 0435.53044
[018] [19] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231-255. | Zbl 0506.53032
[019] [20] K. Yano and M. Kon, Generic submanifolds of Sasakian manifolds, Kodai Math. J. 3 (1980), 163-196. | Zbl 0452.53034
[020] [21] K. Yano and M. Kon, Structures on Manifolds, World Sci., Singapore, 1984.