Warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold

We introduce warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold. We give a necessary and suﬃcient condition for a skew semi-invariant submanifold of order 1 to be a locally warped product. We also establish an inequality between the warping function and the squared norm of the second fundamental form for such submanifolds. The equality case is also discussed.

PDF

___

[1] Adati T. Submanifolds of an almost product manifold. Kodai Math J 1981; 4: 327343.
[2] Al-Solamy FR, Khan MA. Warped product submanifolds of Riemannian product manifolds. Abstract and Applied Analysis 2012; Article ID 724898, 12 pages, doi:10.1155/2012/724898.
[3] Arslan K, Carriazo A, Chen BY, Murathan C. On slant submanifolds of neutral Kaehler manifolds. Taiwanese J Math 2010; 17: 561584.
[4] At¸ceken M. Warped product semi-slant submanifolds in locally Riemannian product manifolds. Bull Austral Math Soc 2008; 77: 177186.
[5] Bejancu A. CR-Submanifolds of Kaehler manifold I. Proc Amer Math Soc 1978; 69: 135142.
[6] Bejancu A. Semi-invariant submanifolds of locally product Riemannian manifolds. An Univ Timi¸soara Ser S¸tiint Math Al 1984; 22: 311.
[7] Bishop RL, ONeill B. Manifolds of negative curvature. Trans Amer Math Soc 1969; 145: 149.
[8] Cabrerizo JL, Carriazo A, Fern´andez LM, Fern´andez M. Slant submanifolds in Sasakian manifolds. Glasgow Math J 2000; 42: 125138.
[9] Carriazo A. Bi-slant immersions. In: Proc ICRAMS 2000, Kharagpur, India, 2000, 8897.
[10] Chen BY. Slant immersions. Bull Austral Math Soc 1990; 41: 135147.
[11] Chen BY. Geometry of warped product in Kaehler manifolds. Monatsh Math 2001; 133: 177195.
[12] Chen BY. Geometry of warped product submanifolds: a survey. J Adv Math Stud 2013; 6: 143.
[13] Dillen F, N¨olker S. Semi-paralellity multi rotation surfaces and the helix property. J Reine Angew Math 1993; 435: 3363.
[14] Hiepko S. Eine innere Kennzeichnung der verzertten Produkte. Math Ann 1979; 241: 209215 (in German).
[15] Khan VA,Khan MA.Pseudo-slant submanifolds of a Sasakian manifold.Indian J Pure Appl Math 2007; 38: 3142.
[16] Khursheed Haider SM, Thakur M, Advin M. Warped product skew CR-submanifolds of a Cosymplectic manifold. Lobachevskii J Math 2012; 33: 262273.
[17] Liu X, Shao FM. Skew semi-invariant submanifolds of a locally product manifold. Portugaliae Math 1999; 56: 319327.
[18] Lotta A. Slant submanifolds in contact geometry. Bull Math Soc Sci Roumanie 1996; 39: 183198.
[19] Papaghiuc N. Semi-slant submanifolds of a K¨ahlerian manifold. Ann S¸t Al I Cuza Univ Ia¸si 1994; 40: 5561.
[20] Ronsse GS.Generic andskew CR-submanifoldsof aK¨ahlermanifold.BullInst MathAcad Sinica 1990; 18: 127141.
[21] S¸ahin B. Slant submanifolds of an almost product Riemannian manifold. J Korean Math Soc 2006; 43: 717732.
[22] S¸ahin B. Warped product semi-invariant submanifolds of a locally product Riemannian manifold. Bull Math Soc Sci Roumanie 2006; 49: 383394.
[23] S¸ahin B. Warped product semi-slant submanifolds of a locally product Riemannian manifold. Studia Sci Math Hungarica 2009; 46: 169-184.
[24] S¸ahin B.Warped product submanifolds of a K¨ahler manifold with a slant factor. Ann Pol Math 2009; 95: 207226.
[25] S¸ahin B. Skew CR-warped product submanifolds of a K¨ahler manifolds. Math Commun 2010; 15: 189204.
[26] Ta¸stan HM, Ozdemir F. The geometry of hemi-slant submanifolds of a locally product Riemannian manifold. Turk J Math 2015; 39: 268284.
[27] Tripathi MM. Generic submanifolds of generalized complex space forms. Publ Math Debrecen 1997: 50: 373392.
[28] Uddin S, Chi AYM. Warped product pseudo-slant submanifolds of nearly Kaehler manifolds. An St Univ Ovidius Constant¸a 2011; 3: 195204.
[29] Yano K, Kon M. Structures on Manifolds. Singapore: World Scientiﬁc, 1984.