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

Bang-Yen Chen

Open Mathematics (2010)

  • Volume: 8, Issue: 4, page 706-734
  • ISSN: 2391-5455

Abstract

top
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 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.

How to cite

top

Bang-Yen Chen. "Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space." Open Mathematics 8.4 (2010): 706-734. <http://eudml.org/doc/269341>.

@article{Bang2010,
abstract = {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.},
author = {Bang-Yen Chen},
journal = {Open Mathematics},
keywords = {Lorentz surface; Parallel surface; Neutral indefinite space form; Pseudo hyperbolic 4-space; submanifold theory; parallel surfaces; indefinite space forms},
language = {eng},
number = {4},
pages = {706-734},
title = {Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space},
url = {http://eudml.org/doc/269341},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Bang-Yen Chen
TI - Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space
JO - Open Mathematics
PY - 2010
VL - 8
IS - 4
SP - 706
EP - 734
AB - 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.
LA - eng
KW - Lorentz surface; Parallel surface; Neutral indefinite space form; Pseudo hyperbolic 4-space; submanifold theory; parallel surfaces; indefinite space forms
UR - http://eudml.org/doc/269341
ER -

References

top
  1. [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. [2] Chen B.Y., Geometry of Submanifolds, Pure and Applied Mathematics, 22, Marcel Dekker, New York, 1973 
  3. [3] Chen B.Y., Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1, World Scientific, Teaneck, 1984 
  4. [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. [5] Chen B.Y., Classification of marginally trapped Lorentzian flat surfaces in 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. [6] Chen B.Y., Marginally trapped surfaces and Kaluza-Klein theory, Int. Electron. J. Geom., 2009, 2(1), 1–16 Zbl1207.83003
  7. [7] Chen B.Y., Black holes, marginally trapped surfaces and quasi-minimal surfaces, Tamkang J. Math., 2009, 40(4), 313–341 Zbl1194.53060
  8. [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. [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) Zbl1217.53061
  10. [10] Chen B.Y., Complete classification of parallel Lorentz surfaces in neutral pseudo 4-sphere, (submitted) 
  11. [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] Zbl1192.53016
  12. [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 Zbl0811.53054
  13. [13] Chen B.Y., Garay O.J., Classification of quasi-minimal surfaces with parallel mean curvature vector in pseudo-Euclidean 4-space 𝔼 2 4 , Results. Math., 2009, 55(1–2), 23–38 http://dx.doi.org/10.1007/s00025-009-0386-9 Zbl1178.53049
  14. [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 Zbl1182.53018
  15. [15] Ferus D., Immersions with parallel second fundamental form, Math. Z., 1974, 140, 87–93 http://dx.doi.org/10.1007/BF01218650[Crossref] Zbl0279.53048
  16. [16] Graves L.K., On codimension one isometric immersions between indefinite space forms, Tsukuba J. Math., 1979, 3(2), 17–29 Zbl0439.53065
  17. [17] Graves L.K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc., 1979, 252, 367–392 Zbl0415.53041
  18. [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] Zbl1165.83334
  19. [19] Magid M.A., Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math., 1984, 8(1), 31–54 Zbl0549.53052
  20. [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. [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] Zbl0125.21206
  22. [22] Strübing W., Symmetric submanifolds of Riemannian manifolds, Math. Ann., 1979, 245(1), 37–44 http://dx.doi.org/10.1007/BF01420428[Crossref] Zbl0424.53025
  23. [23] Takeuchi M., Parallel submanifolds of space forms, In: Manifolds and Lie Groups, Progr. Math., 14, Birkhäuser, Boston, 1981, 429–447 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.