Abstract
O’Neill introduced the notion of Riemannian submersions (cf. O’Neill, Mich. Math. J. 13, 459–469, 1966, [28]). For the submersion \(\pi :M\longrightarrow N\) of a CR submanifold M of a Kaehler manifold \(\bar{M}\) onto an almost Hermitian manifold N, Kobayashi (cf. Kobayashi, Tohoku Math. J. 39, 95–100, 1987, [21]) proved that N becomes a Kaehler manifold. In this article, we briefly summarize the contributions on submersions of CR submanifolds of some almost Hermitian manifolds and almost contact metric manifolds.
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References
Ali, S., Hussain, S.I.: Submersions of CR submanifolds of a nearly-Kaehler manifold-I,II. Radovi Matematicki, 7(1991), 197–205: 8(1992), 281–289
Baird, P., Wood, J.C.: Harmonic Morphisms Between Riemannian Manifolds. CLarendon Press, Oxford (2003)
Bejancu, A.: CR submanifolds of a Kaehler manifold I, II. Proc. Am. Math. Soc. 69, 135–142 (1978); Trans. Am. Math. Soc. 250, 333–345 (1979)
Bejancu, A.: Geometry of CR Submanifolds. D. Reidel Publishing Company, Dordrecht (1986)
Blair, D.E., Chen, B.Y.: On CR submanifolds of Hermitian manifolds. Israel J. Math. 34, 353–363 (1979)
Blair, D.E., Dragomir, S.: CR products in locally conformal Kahler manifolds. Kyushu J. Math. 56, 337–362 (2002)
Chen, B.Y.: CR submanifolds of a Kaehler manifold-I, II. J. Differ. Geom. 16 305–322, 493–509 (1981)
Chen, B.Y.: Geometry of Submanifolds and its Applications. Science University of Tokyo, Tokyo (1981)
Chen, B.Y.: Cohomology of CR-submanifolds. Ann. Fac. Sci. Toulouse 3(2), 167–172 (1981)
Chen, B.Y., Vanhecke, L.: Isometric, holomorphic and symplectic reflections. Geom. Dedicata 29, 259–277 (1989)
Chinea, D.: Almost contact metric submersions. Rend. Circ. Mat. Palermo 34, 89–104 (1985)
Chinea, D., Gonzalez, C.: A classification of almost contact metric manifolds. Ann. Mat. Pura Appl. 156, 15–36 (1990)
Deshmukh, S., Ali, S., Hussain, S.I.: Submersions of CR submanifolds of a Kaehler manifold. Indian J. Pure Appl. Math. 19(12), 1185–1205 (1988)
Deshmukh, S., Ghazal, T., Hashem, H.: Submersions of CR submanifolds on an almost Hermitian manifold. Yokohama Math. J. 40, 45–57 (1992)
Dragomir, S.: Cauchy-Riemann submanifolds of locally conformal Kaehler manifolds I-II. Geom. Dedicata, 28, 181–197 (1988). Atti Sem. Mat. Fis. Univ. Modena 37, 1–11 (1989)
Falcitelli, M., Ianus, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific, Singapore (2004)
Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967)
Gudmundsson, S.: The Geometry of Harmonic Morphisms, Ph.D. thesis, University of Leeds (1992)
Ianus, S., Mazzocco, R., Vilcu, G.E.: Riemannian submersions from quaternionic manifold. Acta Appl. Math. 104, 83–89 (2008)
Ianus, S., Marchiafava, S., Vilcu, G.E.: Para-quaternionic CR submanifolds of para-quaternionic Kaehler manifold and semi-Riemannian submersions. Cent. Eur. J. Math. 8, 735–753 (2010)
Kobayashi, S.: Submersions of CR submanifolds. Tohoku Math. J. 39, 95–100 (1987)
Mangione, V.: Some submersions of CR hypersurfaces of Kaehler-Einstein manifold. Int. J. Math. Math. Sci. 18, 1137–1144 (2003)
Marrero, J.C., Rocha, J.: Locally conformal Kaehler submersions. Geom. Dedicata 52, 271–289 (1994)
Massamba, F., Matamba, T.T.: Horizontally submersions of contact CR submanifolds. Turkish J. Math. 38, 436–453 (2014)
Matsumoto, K.: On CR submanifolds of locally conformal Kaehler manifold. J. Korean Math. Soc. 21, 49–61 (1984)
Narita, F.: CR submanifolds of locally conformal Kaehler manifolds and Riemannian submersions. Colloq. Math. 70, 165–179 (1996)
Nomizu, K., Yano, K.: On circles and spheres in Riemannian geometry. Math. Ann. 210, 163–170 (1974)
O’Neill, B.: The fundamental equations of submersions. Mich. Math. J. 13, 459–469 (1966)
Papaghiuc, N.: Submersions of semi-invariant submanifolds of a Sasakian manifold. An. St. Univ. Al. I. Cuza Iasi, Matematica 35, 281–288 (1989)
Sahin, B.: Horizontally conformal submersions of CR submanifolds. Kodai Math. J. 31, 46–53 (2008)
Shahid, M.H., Solamy, F.R.: Submersions of quaternion CR submanifolds of a quaternion Kaehler manifold. Math. J. Toyama Univ. 23, 93–113 (2000)
Vaisman, I.: On locally conformal almost Kaehler manifolds. Israel J. Math. 24, 338–351 (1976)
Vaisman, I.: Locally conformal Kaehler manifolds with parallel Lee form. Rend. Mat. 12, 263–284 (1979)
Watson, B.: Almost Hermitian submersions. J. Differ. Geom. 11, 147–165 (1970)
Yano, K., Kon, M.: Structures on Manifolds. World Scientific, Singapore (1984)
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Shahid, M.H., Al-Solamy, F.R., Jamali, M. (2016). Submersions of CR Submanifolds. In: Dragomir, S., Shahid, M., Al-Solamy, F. (eds) Geometry of Cauchy-Riemann Submanifolds. Springer, Singapore. https://doi.org/10.1007/978-981-10-0916-7_11
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