PARALLEL AND SEMIPARALLEL LIGHTLIKE HYPERSURFACES OF SEMI-RIEMANNIAN SPACE FORMS

In this paper, some properties of lightlike hypersurfaces with parallel and semiparallel second fundamental forms are investigated in semi-Riemannian space forms. Then some generalizations of these conditions are performed.

Keywords:

Lightlike hypersurfaces, parallel, semiparallel, semi-Riemannian space forms, 2-parallel 2-semiparallel,

PDF

___

[1] S. Akiba, Submanifolds with zat normal connection and parallel second fundamental tensor, Sci. Repts Yokohama Nat. Univ. Sec. I, 23 (1976), 7-14.
[2] J. Deprez, Semi-parallel surfaces in Euclidean space, J. Geom., 25 (1985), 192-200.
[3] J. Deprez, Semi-parallel hypersurfaces, Rend. Semin. Mat. Univ. Politec. Torino, 44 (1986), 303-316.
[4] F. Dillen, The classi. . . cation of hypersurfaces of a Euclidean space with parallel higher order fundamental form, Math. Z., 203 (1990), 635-643.
[5] F. Dillen, Hypersurfaces of a real space form with parallel higher order fundamental form, Soochow J. Math., 18 (1992), 321-338.
[6] F. Dillen, Semi-parallel hypersurfaces of a real space form, Israel J. Math., 75 (1991), 193-202.
[7] Duggal, K.L. and Bejancu, A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academics Publishers,1996.
[8] Duggal, K.L. and Jin, D.H., A classi cation of Einstein lightlike hypersurfaces of a Lorentzian space form, J. Geom. Phys., 60 (2010), 1881-1889.
[9] Duggal, K.L. and Sahin, B., Dierential Geometry of Lightlike Submanifolds, Birkhauser Verlag AG, 2010.
[10] Gunes, R., Sahin, B. ve Klc, E., On Lightlike Hypersurfaces of a Semi-Riemannian Space Form, Turk. J. Math., 27 (2003), 283-297.
[11] U. Lumiste, Semiparallel Submanifolds in Space Forms, Springer, 2009.
[12] S. Maeda, Isotropic immersions with parallel second fundamental form, Canad. Math. Bull., 26 (1983), 291-296.
[13] M. A. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math., 8 (1984), 31-54.
[14] V. Mirzoyan, On submanifolds with parallel second fundamental form in spaces of constant curvature, Tartu Ulik. Toim. Acta Comm. Univ. Tartuensis, 464 (1978), 59-74 (in Russian; summary in English).
[15] V. Mirzoyan, On submanifolds with parallel fundamental form of higher order, Dokl. Akad. Nauk Armenian SSR, 66 (1978), 71-75 (in Russian).
[16] H. Naitoh, Isotropic submanifolds with parallel second fundamental forms in symmetric spaces, Osaka J. Math., 17 (1980), 95-100.
[17] Peterson, P., Riemannian Geometry 2nd Ed., Springer, 2006.
[18] Sahin, B., Lightlike Hypersurfaces of Semi-Euclidean Spaces Satisfying Curvature Conditions of Semisymmetry Type, Turk. J. Math., 31 (2007), 139-162.
[19] U. Simon and A. Weinstein, Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flachentheorie, Manuscripta Math., 1 (1969), 139-146.
[20] M. Takeuchi, Parallel submanifolds of space forms, in Manifolds and Lie Groups: Papers in Honor of Y. Matsushima, Birkhauser, Basel, (1981), 429-447.
[21] J. Vilms, Submanifolds of Euclidean space with parallel second fundamental form, Proc. Amer. Math. Soc., 32 (1972), 263-267.