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2009 | 7 | 3 | 400-428
Tytuł artykułu

Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Submanifolds with parallel mean curvature vector play important roles in differential geometry, theory of harmonic maps as well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean curvature vector were classified by B. Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean curvature vector in pseudo-Riemannian spheres and pseudo-hyperbolic spaces with arbitrary codimension and arbitrary index. Consequently, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in all pseudo-Riemannian space forms. As an immediate by-product, we obtain the complete classifications of spatial surfaces with parallel mean curvature vector in arbitrary Lorentzian space forms.
Wydawca
Czasopismo
Rocznik
Tom
7
Numer
3
Strony
400-428
Opis fizyczny
Daty
wydano
2009-09-01
online
2009-08-12
Twórcy
Bibliografia
  • [1] Chen B.Y., Geometry of submanifolds, M. Dekker, New York, 1973
  • [2] Chen B.Y., On the surface with parallel mean curvature vector, Indiana Univ. Math. J., 1973, 22, 655–666 http://dx.doi.org/10.1512/iumj.1973.22.22053
  • [3] Chen B.Y., Total mean curvature and submanifolds of finite type, World Scientific, New Jersey, 1984
  • [4] Chen B.Y., Finite type submanifolds in pseudo-Euclidean spaces and applications, Kodai Math. J., 1985, 8, 358–374 http://dx.doi.org/10.2996/kmj/1138037104
  • [5] Chen B.Y., Riemannian submanifolds, Handbook of differential geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000
  • [6] Chen B.Y., Marginally trapped surfaces and Kaluza-Klein theory, Intern. Elect. J. Geom., 2009, 2, 1–16
  • [7] Chen B.Y., Classification of spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension, J. Math. Phys., 2009, 50, 043503, 14 pages http://dx.doi.org/10.1063/1.3100755
  • [8] Chen B.Y., Van der Veken J., Spatial and Lorentzian surfaces in Robertson-Walker space-times, J. Math. Phys., 2007, 48, 073509, 12 pages http://dx.doi.org/10.1063/1.2748616
  • [9] Chen B.Y., Van der Veken J., Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms, Tohoku Math. J., 2009, 61, 1–40 http://dx.doi.org/10.2748/tmj/1238764545
  • [10] Hawking S.W., Penrose R., The singularities of gravitational collapse and cosmology, Proc. Roy. Soc. London Ser. A, 1970, 314, 529–548 http://dx.doi.org/10.1098/rspa.1970.0021
  • [11] Magid M.A., Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math., 1984, 8, 31–54
  • [12] O’Neill B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1982
  • [13] Penrose R., Gravitational collapse and space-time singularities, Phys. Rev. Lett., 1965, 14, 57–59 http://dx.doi.org/10.1103/PhysRevLett.14.57
  • [14] Takahashi T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 1966, 18, 380–385 http://dx.doi.org/10.2969/jmsj/01840380
  • [15] Yau S.T., Submanifolds with constant mean curvature I, Amer. J. Math., 1974, 96, 346–366 http://dx.doi.org/10.2307/2373638
  • [16] Verstraelen L., Pieters M., Some immersions of Lorentz surfaces into a pseudo-Riemannian space of constant curvature and of signature (2; 2), Rev. Roumaine Math. Pures Appl., 1976, 21, 593–600
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-009-0020-9
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