Classification of spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension. (English) Zbl 1214.53021
Summary: Spatial surfaces with parallel mean curvature vector play some important roles in general relativity, theory of harmonic maps, as well as in differential geometry. Recently, spatial surfaces in four-dimensional Minkowski space time with parallel mean curvature vector were classified by the author and J. Van der Veken [Tôhoku Math. J. (2) 61, No. 1, 1–40 (2009; Zbl 1182.53018)]. In this article, we completely classify spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension. The main result states that there exist 16 families of such surfaces. As by-product, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in Minkowski space times of arbitrary dimension.
Editorial remark: No review copy delivered
Editorial remark: No review copy delivered
MSC:
| 53C10 | \(G\)-structures |
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
Citations:
Zbl 1182.53018References:
| [1] | DOI: 10.1007/BF01456410 · Zbl 0476.53013 · doi:10.1007/BF01456410 |
| [2] | DOI: 10.1007/BF02764073 · Zbl 1038.58011 · doi:10.1007/BF02764073 |
| [3] | Chen B. Y., Geometry of Submanifolds (1973) · Zbl 0262.53036 |
| [4] | DOI: 10.1142/0065 · doi:10.1142/0065 |
| [5] | Chen B. Y., Soochow J. Math. 22 pp 117– (1996) |
| [6] | Chen, B. Y. Riemannian submanifolds, Handbook of Differential Geometry (North-Holland, Amsterdam, 2000), Vol. I, pp. 187–418. |
| [7] | Chen B. Y., C. R. Acad. Sci. Paris 317 pp 961– (1993) |
| [8] | Chen B. Y., Memoirs Fac. Sci. Kyushu Univ. Ser. A, Math. 45 pp 325– (1991) |
| [9] | DOI: 10.1088/0264-9381/24/3/003 · Zbl 1141.53065 · doi:10.1088/0264-9381/24/3/003 |
| [10] | DOI: 10.1063/1.2748616 · Zbl 1144.81324 · doi:10.1063/1.2748616 |
| [11] | DOI: 10.2748/tmj/1238764545 · Zbl 1182.53018 · doi:10.2748/tmj/1238764545 |
| [12] | DOI: 10.1016/j.jmaa.2005.05.049 · Zbl 1091.53038 · doi:10.1016/j.jmaa.2005.05.049 |
| [13] | DOI: 10.1088/0264-9381/24/22/009 · Zbl 1165.83334 · doi:10.1088/0264-9381/24/22/009 |
| [14] | DOI: 10.1017/CBO9780511524646 · Zbl 0265.53054 · doi:10.1017/CBO9780511524646 |
| [15] | DOI: 10.1098/rspa.1970.0021 · Zbl 0954.83012 · doi:10.1098/rspa.1970.0021 |
| [16] | DOI: 10.1090/S0002-9904-1972-12942-2 · Zbl 0236.53054 · doi:10.1090/S0002-9904-1972-12942-2 |
| [17] | DOI: 10.1007/BF00762194 · Zbl 0537.53064 · doi:10.1007/BF00762194 |
| [18] | DOI: 10.1007/BF01297766 · Zbl 0866.53011 · doi:10.1007/BF01297766 |
| [19] | Magid M. A., Tsukuba J. Math. 8 pp 31– (1984) |
| [20] | O’Neill B., Semi-Riemannian Geometry with Applications to Relativity (1982) |
| [21] | DOI: 10.1103/PhysRevLett.14.57 · Zbl 0125.21206 · doi:10.1103/PhysRevLett.14.57 |
| [22] | Penrose R., Black Holes and Relativistic Stars pp 103– (1998) |
| [23] | Verstraelen L., Rev. Roum. Math. Pures Appl. 21 pp 593– (1976) |
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