Paraquaternionic CR-submanifolds of paraquaternionic Kähler manifolds and semi-Riemannian submersions
Stere Ianuş ; Stefano Marchiafava ; Gabriel Vîlcu
Open Mathematics, Tome 8 (2010), p. 735-753 / Harvested from The Polish Digital Mathematics Library

In this paper we introduce paraquaternionic CR-submanifolds of almost paraquaternionic hermitian manifolds and state some basic results on their differential geometry. We also study a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of paraquaternionic Kähler manifolds.

Publié le : 2010-01-01
EUDML-ID : urn:eudml:doc:269045
@article{bwmeta1.element.doi-10_2478_s11533-010-0048-x,
     author = {Stere Ianu\c s and Stefano Marchiafava and Gabriel V\^\i lcu},
     title = {Paraquaternionic CR-submanifolds of paraquaternionic K\"ahler manifolds and semi-Riemannian submersions},
     journal = {Open Mathematics},
     volume = {8},
     year = {2010},
     pages = {735-753},
     zbl = {1202.53054},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0048-x}
}
Stere Ianuş; Stefano Marchiafava; Gabriel Vîlcu. Paraquaternionic CR-submanifolds of paraquaternionic Kähler manifolds and semi-Riemannian submersions. Open Mathematics, Tome 8 (2010) pp. 735-753. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0048-x/

[1] Al-Aqeel A., Bejancu A., On normal semi-invariant submanifolds of paraquaternionic Kähler manifolds, Toyama Math. J., 2007, 30, 63–75 | Zbl 1146.53011

[2] Alekseevsky D., Cortés V., The twistor spaces of a para-quaternionic Kähler manifold, Osaka J. Math., 2008, 45(1), 215–251 | Zbl 1177.53047

[3] Alekseevsky D., Kamishima Y., Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds, Cent. Eur. J. Math., 2004, 2(5), 732–753 http://dx.doi.org/10.2478/BF02475974[Crossref] | Zbl 1116.53046

[4] Alekseevsky D.V., Marchiafava S., Almost complex submanifolds of quaternionic manifolds, In: Proceedings of the colloquium on differential geometry, Debrecen (Hungary), 25–30 July 2000, Inst. Math. Inform. Debrecen, 2001, 23–38 | Zbl 1075.53529

[5] Barros M., Chen B.-Y., Urbano F., Quaternion CR-submanifolds of quaternion manifolds, Kodai Math. J., 1981, 4(3), 399–417 http://dx.doi.org/10.2996/kmj/1138036425[Crossref] | Zbl 0481.53046

[6] Bejancu A., CR submanifolds of a Kaehler manifold I, Proc. Am. Math. Soc., 1978, 69(1), 135–142 | Zbl 0368.53040

[7] Bejancu A., Geometry of CR-submanifolds, Mathematics and its Applications (East European Series), 23, D. Reidel Publishing Co., Dordrecht, 1986 | Zbl 0605.53001

[8] Bejancu A., Normal semi-invariant submanifolds of paraquaternionic Kähler manifolds, Kuwait J. Sci. Engrg., 2006, 33(2), 33–45 | Zbl 1207.53063

[9] Bejancu A., Farran H.R., Foliations and Geometric Structures, Mathematics and Its Applications, 580, Springer, Dordrecht, 2006 | Zbl 1092.53021

[10] Blažić N., Paraquaternionic projective spaces and pseudo-Riemannian geometry, Publ. Inst. Math. (Beograd), 1996, 60(74), 101–107 | Zbl 1002.53020

[11] Blažić N., Vukmirović S., Solutions of Yang-Mills equations on generalized Hopf bundles, J. Geom. Phys., 2002, 41(1–2), 57–64 [Crossref] | Zbl 1009.53021

[12] Chen B.-Y., Cohomology of CR-submanifolds, Ann. Fac. Sci. Toulouse Math., 1981, 3(2), 167–172 [Crossref] | Zbl 0478.53046

[13] Dancer A.S., Jørgensen H.R., Swann A.F., Metric geometries over the split quaternions, Rend. Sem. Mat. Univ. Politec. Torino, 2005, 63(2), 119–139 | Zbl 1178.53041

[14] Falcitelli M., Ianuş S., Pastore A.M., Riemannian Submersions and Related Topics, World Scientific, River Edge, 2004 http://dx.doi.org/10.1142/9789812562333[Crossref] | Zbl 1067.53016

[15] Galicki K., Lawson H.B., Quaternionic reduction and quaternionic orbifolds, Math. Ann., 1988, 282(1), 1–21 http://dx.doi.org/10.1007/BF01457009[Crossref] | Zbl 0628.53060

[16] Garcia-Rio E., Matsushita Y., Vásquez-Lorenzo R., Paraquaternionic Kähler manifolds, Rocky Mountain J. Math., 2001, 31(1), 237–260 http://dx.doi.org/10.1216/rmjm/1008959679[Crossref] | Zbl 0987.53020

[17] Ianuş S., Ionescu A.M., Vîlcu G.E., Foliations on quaternion CR-submanifolds, Houston J. Math., 2008, 34(3), 739–751 | Zbl 1155.53017

[18] Ianuş S., Mazzocco R., Vîlcu G.E., Real lightlike hypersurfaces of paraquaternionic Kähler manifolds, Mediterr. J. Math., 2006, 3(3–4), 581–592 [Crossref] | Zbl 1167.53307

[19] Ianuş S., Vîlcu G.E., Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles, Int. J. Geom. Methods Mod. Phys., 2008, 5(6), 893–903 http://dx.doi.org/10.1142/S0219887808003016[Crossref][WoS] | Zbl 1156.53044

[20] Ionescu A.M., Vîlcu G.E., A note on paraquaternionic manifolds, Missouri J. Math. Sci., 2007, 19(3), 213–218 | Zbl 1152.53024

[21] Ivanov S., Zamkovoy S., Parahermitian and paraquaternionic manifolds, Differential Geom. Appl., 2005, 23(2), 205–234 http://dx.doi.org/10.1016/j.difgeo.2005.06.002[Crossref]

[22] Kobayashi S., Submersions of CR submanifolds, Tôhoku Math. J., 1987, 39(1), 95–100 | Zbl 0619.58004

[23] Libermann P., Sur le problème d’équivalence de certaines structures infinitésimales, Ann. Mat. Pura Appl., 1954, 36, 27–120 http://dx.doi.org/10.1007/BF02412833[Crossref] | Zbl 0056.15401

[24] Marchiafava S., Submanifolds of (para)-quaternionic Kähler manifolds, Note Mat., 2008, 28(suppl.1), 295–316

[25] O’Neill B., Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, New York, 1983

[26] Ornea L., CR-submanifolds. A class of examples, Rev. Roumaine Math. Pures Appl., 2006, 51(1), 77–85 | Zbl 1119.53313

[27] Vaccaro M., Kaehler and para-Kaehler submanifolds of a para-quaternionic Kaehler manifold, Ph.D. thesis, Università degli Studi di Roma II’ Tor Vergata’, Rome, Italy, 2007

[28] Vîlcu G.E., Submanifolds of an Almost Paraquaternionic Kähler Product Manifold, Int. Math. Forum, 2007, 2(15), 735–746 | Zbl 1156.53308

[29] Vukmirovic S., Paraquaternionic reduction, preprint available at http://arxiv.org/abs/math/0304424

[30] Wu H., On the de Rham decomposition theorem, Illinois J. Math., 1964, 8, 291–311 | Zbl 0122.40005

[31] Yano K., Kon M., CR Submanifolds of Kaehlerian and Sasakian Manifolds, Progress in Mathematics, 30, Birkhäuser, Boston, 1983

[32] Zamkovoy S., Geometry of paraquaternionic Kähler manifolds with torsion, J. Geom. Phys., 2006, 57(1), 69–87 http://dx.doi.org/10.1016/j.geomphys.2006.02.003[Crossref] | Zbl 1109.53035