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

Bang-Yen Chen

Open Mathematics (2009)

  • Volume: 7, Issue: 3, page 400-428
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Bang-Yen Chen. "Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms." Open Mathematics 7.3 (2009): 400-428. <http://eudml.org/doc/269711>.

@article{Bang2009,
abstract = {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.},
author = {Bang-Yen Chen},
journal = {Open Mathematics},
keywords = {Spatial surface; Parallel mean curvature vector; Lorentzian space form; Pseudo-Riemannian space form; CMC surface; Light cone; pseudo-Riemannian space form; parallel mean curvature vector; isometric immersions},
language = {eng},
number = {3},
pages = {400-428},
title = {Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms},
url = {http://eudml.org/doc/269711},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Bang-Yen Chen
TI - Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms
JO - Open Mathematics
PY - 2009
VL - 7
IS - 3
SP - 400
EP - 428
AB - 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.
LA - eng
KW - Spatial surface; Parallel mean curvature vector; Lorentzian space form; Pseudo-Riemannian space form; CMC surface; Light cone; pseudo-Riemannian space form; parallel mean curvature vector; isometric immersions
UR - http://eudml.org/doc/269711
ER -

References

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  1. [1] Chen B.Y., Geometry of submanifolds, M. Dekker, New York, 1973 
  2. [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 Zbl0252.53021
  3. [3] Chen B.Y., Total mean curvature and submanifolds of finite type, World Scientific, New Jersey, 1984 Zbl0537.53049
  4. [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 Zbl0586.53022
  5. [5] Chen B.Y., Riemannian submanifolds, Handbook of differential geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000 
  6. [6] Chen B.Y., Marginally trapped surfaces and Kaluza-Klein theory, Intern. Elect. J. Geom., 2009, 2, 1–16 Zbl1207.83003
  7. [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 Zbl1214.53021
  8. [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 Zbl1144.81324
  9. [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 Zbl1182.53018
  10. [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 Zbl0954.83012
  11. [11] Magid M.A., Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math., 1984, 8, 31–54 Zbl0549.53052
  12. [12] O’Neill B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1982 
  13. [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 Zbl0125.21206
  14. [14] Takahashi T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 1966, 18, 380–385 http://dx.doi.org/10.2969/jmsj/01840380 Zbl0145.18601
  15. [15] Yau S.T., Submanifolds with constant mean curvature I, Amer. J. Math., 1974, 96, 346–366 http://dx.doi.org/10.2307/2373638 Zbl0304.53041
  16. [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 Zbl0333.53017

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