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

Open Mathematics (2009)

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

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topBang-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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- [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
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