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A report on canonical null curves and screen distributions for lightlike geometry. (English) Zbl 1117.53019
The paper under review is a survey paper which can serve as a reference to researchers in light-like geometry and also can stimulate further research on finding more cases of canonical null curves and screens for light-like geometry. The general theory of light-like submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The main goal of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. The author also proves a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.
##### MSC:
 53B25 Local submanifolds 53C50 Lorentz manifolds, manifolds with indefinite metrics 53B50 Applications of local differential geometry to physics
##### References:
 [1] Akivis, M.A., Goldberg, V.V.: On some methods of construction of invariant normalizations of lightlike hypersurfaces. Differential Geom. Appl. 12(2), 121–143 (2000) · Zbl 0965.53022 · doi:10.1016/S0926-2245(00)00008-5 [2] Atindogbe, C., Duggal, K.L.: Conformal Screen on Lightlike Hypersurfaces. Int. J. Pure Appl. Math. 11(4), 421–442 (2004) [3] Beem, J.K., Ehrlich, P.E.: Global Lorentzian Geometry. Marcel Dekker, New York (1981) (Second edn. with Easley, K. L. 1996) [4] Bejancu, A.: Geometry of CR-submanifolds. Reidel, Amsterdam, The Netherlands (1986) [5] Bejancu, A.: A canonical screen distribution on a degenerate hypersurface. Sci. Bull. Ser. A, Appl. Math. Phys. 55, 55–61 (1993) [6] Bejancu, A.: Lightlike curves in Lorentz manifolds. Publ. Math. Debrecen 44(f.1-2), 145–155 (1994) [7] Bejancu, A., Duggal, K.L.: Degenerate hypersurfaces of semi-Riemannian manifolds. Bul. Inst. Politeh. Iasi 37, 13–22 (1991) [8] Bejancu, A., Duggal, K.L.: Lightlike submanifolds of semi-Riemannian manifolds and applications. Kluwer, Dordrecht 364 (1996) [9] Bejancu, A., Ferrández, A., Lucas P.: A new viewpoint on geometry of a lightlike hypersurface in a semi-Euclidean space. Saitama Math. J. 31–38 (1998) [10] Bonnor, W.B.: Null curves in a Minkowski spacetime. Tensor (N.S.) 20, 229–242 (1996) [11] Bonnor, W.B.: Null hypersurfaces in Minkowski spacetime. Tensor (N.S.) 24, 329–345 (1972) [12] Cartan, E.: La Theorie Desk Groupes Finis et Continus et la Geometrie Differentielle. Gauthier-Villars, Paris (1937) [13] Chen, B.Y.: Geometry of Submanifolds. Marcel Dekker, New York (1973) [14] Cöken, A.C., Ciftci, Ü.: On the Cartan curvatures of a null curve in Minkowski spacetime. Geom. Dedicata 114, 71–78 (2005) [15] Duggal, K.L.: Lorentzian geometry of CR-submanifolds. Acta. Appl. Math. 17, 171–193 (1989) · Zbl 0692.53025 · doi:10.1007/BF00046823 [16] Duggal, K.L.: On scalar curvature in lightlike geometry. J. Geom. Phys. 57, 473–481 (2007) · Zbl 1107.53047 · doi:10.1016/j.geomphys.2006.04.001 [17] Duggal, K.L., Giménez, A.: Lightlike hypersurfaces of Lorentzian manifolds with distinguished screen. J. Geom. Phys. 55, 107–122 (2005) [18] Duggal, K.L., Jin, D.H.: Geometry of null curves. Math. J. Toyama Univ. 22, 95–120 (1999) [19] Duggal, K.L., Jin, D.H.: Half lightlike submanifolds of codimension 2. Math. J. Toyama Univ. 22, 121–161 (1999) [20] Duggal, K.L., Jin, D.H.: Null curves and hypersurfaces of semi-Riemannian manifolds. World Scientific (2007) (in press) [21] Duggal, K.L., Sahin, B.: Screen conformal half-lightlike submanifolds. Int. J. Math. & Math. Sci. 68, 3737–3753 (2004) · Zbl 1071.53041 · doi:10.1155/S0161171204403342 [22] Ferrández, A., Giménez, A., Lucas P.: Null helices in Lorentzian space forms. Internat J. Modern Phys. A16, 4845–4863 (2001) [23] Ferrández, A., Giménez, A., Lucas P.: Degenerate curves in pseudo-Euclidean spaces of index 2, 3rd International Conference on Geometry, Integrability and Quantization, pp.209-223. Coral Press, Sofia (2001) [24] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic, New York (1983) [25] Jin, D.H.: Null curves in Lorentz manifolds J. of Dongguk Univ., vol. 18, pp. 203–212 (1999) [26] Jin, D.H.: Fundamental theorem of null curve. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 7(2), 115–127 (2000) [27] Jin, D.H.: Fundamental theorem for lightlike curves. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10(1), 13–23 (2001) [28] Jin, D.H.: Frenet equations of null curves. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10(2), 71–102 (2003) [29] Jin, D.H.: Natural Frenet equations of null curves. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 12(3), 71–102 (2005) [30] Kupeli, D.N.: Singular semi-Riemannian geometry. Kluwer, Dordrecht, vol. 366, (1996) [31] Leistner, T.: Screen bundles of Lorentzian manifolds and some generalizations of pp-waves. J. Geom. Phys. 56(10), 2117–2134 (2006) · Zbl 1111.53020 · doi:10.1016/j.geomphys.2005.11.010 [32] Lucas, P.: Collection of papers (with his collaborators) on null curves and applications to physics: http://www.um.es/docencia/plucas/ [33] Perlick, V.: On totally umbilical submanifolds of semi-Riemannian manifolds. Nonlinear Anal. 63, 511–518 (2005) · Zbl 1159.53342 · doi:10.1016/j.na.2004.12.033 [34] Sahin, B., Kilic, E., Günes, R.: Null helices in ${𝐑}_{\mathbf{\text{1}}}^{\mathbf{\text{3}}}$ . Differ. Geom.-Dyn. Syst. 3(2), 31–36 (2001) [35] Sharma, R.: CR-submanifolds of semi-Riemannian manifolds with applications to relativity and hydrodynamics. Ph.D. Thesis, University of Windsor, Windsor, Canada (1986) [36] Yano, K., Kon, M.: CR-Submanifolds of Kählerian and Sasakian Manifolds. Birkhauser (1983)