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2010 | 8 | 4 | 706-734
Tytuł artykułu

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

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Abstrakty
EN
A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $$ \mathbb{E}_2^4 $$ and in neutral pseudo 4-sphere S 24 (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H 24 (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H 24 (−1). Conversely, every parallel Lorentz surface in H 24 (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.
Twórcy
  • Department of Mathematics, Michigan State University, East Lansing, Michigan, USA, bychen@math.msu.edu
Bibliografia
  • [1] Blomstrom C., Symmetric immersions in pseudo-Riemannian space forms, In: Global Differential Geometry and Global Analysis, Lecture Notes in Math., 1156, Springer, Berlin, 1985, 30–45 http://dx.doi.org/10.1007/BFb0075084[Crossref]
  • [2] Chen B.Y., Geometry of Submanifolds, Pure and Applied Mathematics, 22, Marcel Dekker, New York, 1973
  • [3] Chen B.Y., Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1, World Scientific, Teaneck, 1984
  • [4] Chen B.Y., Riemannian submanifolds, In: Handbook of Differential Geometry, Vol. I, North-Holland, Amsterdam, 2000, 187–418 http://dx.doi.org/10.1016/S1874-5741(00)80006-0[Crossref]
  • [5] Chen B.Y., Classification of marginally trapped Lorentzian flat surfaces in \( \mathbb{E}_2^4 \) and its application to biharmonic surfaces, J. Math. Anal. Appl., 2008, 340(2), 861–875 http://dx.doi.org/10.1016/j.jmaa.2007.09.021[WoS][Crossref]
  • [6] Chen B.Y., Marginally trapped surfaces and Kaluza-Klein theory, Int. Electron. J. Geom., 2009, 2(1), 1–16
  • [7] Chen B.Y., Black holes, marginally trapped surfaces and quasi-minimal surfaces, Tamkang J. Math., 2009, 40(4), 313–341
  • [8] Chen B.Y., Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension, J. Geom. Phys., 2010, 60(2), 260–280 http://dx.doi.org/10.1016/j.geomphys.2009.09.012[WoS][Crossref]
  • [9] Chen B.Y., Explicit classification of parallel Lorentz surfaces in 4D indefinite space forms with index 3, Bull. Inst. Math. Acad. Sinica (N.S.), (in press)
  • [10] Chen B.Y., Complete classification of parallel Lorentz surfaces in neutral pseudo 4-sphere, (submitted)
  • [11] Chen B.Y., Dillen F., Van der Veken J., Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms, Intern. J. Math., 2010, 21(5), 665–686 http://dx.doi.org/10.1142/S0129167X10006276[Crossref]
  • [12] Chen B.Y., Dillen F., Verstraelen L., Vrancken L., A variational minimal principle characterizes submanifolds of finite type, C. R. Acad. Sci. Paris Sér. I Math., 1993, 317(10), 961–965
  • [13] Chen B.Y., Garay O.J., Classification of quasi-minimal surfaces with parallel mean curvature vector in pseudo-Euclidean 4-space \( \mathbb{E}_2^4 \) , Results. Math., 2009, 55(1–2), 23–38 http://dx.doi.org/10.1007/s00025-009-0386-9
  • [14] Chen B.Y., Van der Veken J., Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms, Tohoku Math. J., 2009, 61(1), 1–40 http://dx.doi.org/10.2748/tmj/1238764545
  • [15] Ferus D., Immersions with parallel second fundamental form, Math. Z., 1974, 140, 87–93 http://dx.doi.org/10.1007/BF01218650[Crossref]
  • [16] Graves L.K., On codimension one isometric immersions between indefinite space forms, Tsukuba J. Math., 1979, 3(2), 17–29
  • [17] Graves L.K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc., 1979, 252, 367–392
  • [18] Haesen S., Ortega M., Boost invariant marginally trapped surfaces in Minkowski 4-space, Classical Quantum Gravity, 2007, 24(22), 5441–5452 http://dx.doi.org/10.1088/0264-9381/24/22/009[WoS]
  • [19] Magid M.A., Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math., 1984, 8(1), 31–54
  • [20] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity. With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, New York, 1983
  • [21] Penrose R., Gravitational collapse and space-time singularities, Phys. Rev. Lett., 1965, 14, 57–59 http://dx.doi.org/10.1103/PhysRevLett.14.57[Crossref]
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  • [23] Takeuchi M., Parallel submanifolds of space forms, In: Manifolds and Lie Groups, Progr. Math., 14, Birkhäuser, Boston, 1981, 429–447
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-010-0044-1
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