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Warped product submanifolds of Lorentzian paracosymplectic manifolds. (English) Zbl 1254.53034

Summary: We study warped product submanifolds of a Lorentzian para-cosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold of the form \(M = M_{\top} \times_f M_{\bot}\) of a Lorentzian para-cosymplectic manifold such that the characteristic vector field is normal to \(M\) is a usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifold are always integrable. A necessary and sufficient condition for a semi-invariant submanifold of a Lorentzian para-cosymplectic manifold to be warped product semi-invariant submanifold is obtained. We also investigate the existence and nonexistence of warped product semi-slant and warped product anti-slant submanifolds in a Lorentzian para-cosymplectic manifold.

MSC:

53B25 Local submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C30 Differential geometry of homogeneous manifolds
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References:

[1] Atçeken M.: Semi-slant submanifolds of an almost product metric manifold. Can. Math. Bull. 53(2), 206-217 (2010) · Zbl 1196.53035 · doi:10.4153/CMB-2010-003-0
[2] Atçeken M.: Warped product semi-slant submanifolds in a locally Riemannian product manifold. Bull. Aust. Math. Soc. 77(2), 177-186 (2008) · Zbl 1152.53022 · doi:10.1017/S0004972708000191
[3] Atçeken M.: Warped product semi-slant submanifolds in Kenmotsu manifolds. Turk. J. Math. 34, 425-432 (2010) · Zbl 1195.53079
[4] Atçeken, M.: Warped product semi-invariant submanifolds in almost paracontact manifolds. Math. Probl. Eng. 2009, Article ID 621625 · Zbl 1184.53023
[5] Bejancu A.: CR-submanifolds of a Kaehler manifold. I. Proc. Am. Math. Soc. 69(1), 135-142 (1978) · Zbl 0368.53040
[6] Bejancu A., Papaghiuc N.: Semi-invariant submanifolds of a Sasakian manifold. Ann. Stiint. Al.I. Cuza Univ. Iasi 27, 163-170 (1981) · Zbl 0535.53042
[7] Bishop R.L., O’Neill, Bishop R.L., O’Neill: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1-49 (1969) · Zbl 0191.52002 · doi:10.1090/S0002-9947-1969-0251664-4
[8] Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhauser Boston, Inc., Boston (2002) · Zbl 1011.53001
[9] Cabrerizo J.L.; Carriazo A.; Fernandez L.M.; Fernandez M.: Semi-slant submanifolds of a Sasakian manifold. Geometriae Dedicate 78, 183-199 (1999) · Zbl 0944.53028 · doi:10.1023/A:1005241320631
[10] CabrerizoJ.L.; CarriazoA.; FernandezL.M.; Fernandez M.: Slant submanifolds in Sasakian manifolds. Glasg. Math. J. 42, 125-138 (2000) · Zbl 0957.53022 · doi:10.1017/S0017089500010156
[11] Cabrerizo J.L., Carriazo A., Fernandez L.M., Fernandez M.: Structure on a slant submanifold of a contact manifold. Indian J. Pure Appl. Math. 31, 857-864 (2000) · Zbl 0984.53034
[12] Chen B.Y.: Slant immersions. Bull. Aust. Math. Soc. 41(1), 135-147 (1990) · Zbl 0677.53060 · doi:10.1017/S0004972700017925
[13] Chen, B.Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Leuven (1990) · Zbl 0716.53006
[14] Chen B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 133, 177-195 (2001) · Zbl 0996.53044 · doi:10.1007/s006050170019
[15] Chen B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds II. Monatsh. Math. 134, 103-119 (2001) · Zbl 0996.53045 · doi:10.1007/s006050170002
[16] De U.C., Sengupta A.K.: CR-submanifolds of a Lorentzian para-Sasakian manifold. Bull. Malays. Math. Sci. Soc. (Second Ser.) 23, 99-106 (2000) · Zbl 1029.53084
[17] De U.C., Shaikh A.A.: Non-existence of proper semi-invariant submanifolds of a Lorentzian para-Sasakian manifold. Bull. Malays. Math. Soc. (Second Ser.) 22, 179-183 (1999) · Zbl 1022.53046
[18] Ehrlich, P.E.: Metric deformations of Ricci and sectional curvature on compact Riemannian manifolds. PhD dissertation, SUNY, Stony Brook, NY (1974) · Zbl 1161.53036
[19] Hasegawa I., Mihai I.: Contact CR-warped product submanifolds in Sasakian manifolds. Geom. Dedicata 102, 143-150 (2003) · Zbl 1066.53103 · doi:10.1023/B:GEOM.0000006582.29685.22
[20] Khan V.A., Khan M.A., Khan K.A.: Slant and semi-slant submanifolds of a Kenmotsu manifold. Math. Slovaca 57(5), 483-494 (2007) · Zbl 1164.53034 · doi:10.2478/s12175-007-0040-5
[21] Khan A.K., Khan V.A., Uddin S.: Warped product submanifolds of cosymplectic manifolds. Balkan J. Geom. Appl. 13(1), 55-65 (2008) · Zbl 1161.53036
[22] Khan M.A., Singh K., Khan V.A.: Slant submanifolds of LP-contact manifolds. Differ. Geom. Dyn. Syst. 12, 102-108 (2010) · Zbl 1200.53015
[23] LiH., Liu X.: Semi-slant submanifolds of a locally Riemannian product manifold. Georgian Math. J. 12, 273-282 (2005) · Zbl 1093.53025
[24] Lotta A.: Slant submanifolds in contact geometry. Bull. Math. Soc. Romanie 39, 183-198 (1996) · Zbl 0885.53058
[25] Mihai, I., Rosca, R.: On Lorentzian P-Sasakian Manifolds. Classical Analysis, pp. 156-169. World Scientific, Singapore (1992) · Zbl 1064.53502
[26] Munteanu, M.I.: Warped product contact CR-submanifolds of Sasakian space forms. Publ. Math. Debrecen 66(1-2), 75-120 (2005) · Zbl 1063.53052
[27] Matsumoto K.: On Lorentzian paracontact manifolds. Bull. Yamagata Univ. Nat. Sci. 12(2), 151-156 (1989) · Zbl 0675.53035
[28] Matsumoto K., Mihai I., Rosca R.: ξ-null geodesic gradient vector fields on a Lorentzian para-Sasakian manifold. J. Korean Math. Soc. 32(1), 17-31 (1995) · Zbl 0828.53042
[29] O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983) · Zbl 0531.53051
[30] Papaghiuc N.: Semi-slant submanifolds of a Kaehlerian manifold. Ann. Stiint. Al.I. Cuza Univ. Iasi 40, 55-61 (1994) · Zbl 0847.53012
[31] Prasad S., Ojha R.H.: Lorentzian paracontact submanifolds. Publ. Math. Debrecen 44, 215-223 (1994) · Zbl 0819.53022
[32] Prasad B.: Semi-invariant submanifolds of a Lorentzian para-Sasakian manifold. Bull. Malays. Math. Soc. (Second Ser.) 21, 21-26 (1998) · Zbl 1021.53044
[33] Sahin B.: Nonexistence of warped product semi-slant submanifolds of Kaehler manifold. Geom. Dedicata 117, 195-202 (2006) · Zbl 1093.53059 · doi:10.1007/s10711-005-9023-2
[34] Sasaki S.: On differentiable manifolds with certain structures which are closely related to almost contact structure I. Tôhoku Math. J. 12(2), 459-476 (1960) · Zbl 0192.27903 · doi:10.2748/tmj/1178244407
[35] Satō I.: On a structure similar to the almost contact structure. Tensor (N.S.) 30(3), 219-224 (1976) · Zbl 0344.53025
[36] Tripathi, M.M.: On semi-invariant submanifolds of Lorentzian almost paracontact manifolds. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 8(1), 1-8 (2001). ISSN 1226-0657 · Zbl 1203.53052
[37] Tripathi M.M.: On semi-invariant submanifolds of LP-cosymplectic manifolds. Bull. Malays. Math. Sci. Soc. (Second Ser.) 24(1), 69-79 (2001) · Zbl 1017.53051
[38] Ünal B.: Doubly warped products. Differ. Geom. Appl. 15, 253-263 (2001) · Zbl 1035.53100 · doi:10.1016/S0926-2245(01)00051-1
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