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http://dx.doi.org/10.4134/JKMS.j150381

ON THE THEORY OF LORENTZ SURFACES WITH PARALLEL NORMALIZED MEAN CURVATURE VECTOR FIELD IN PSEUDO-EUCLIDEAN 4-SPACE  

Aleksieva, Yana (Faculty of Mathematics and Informatics Sofia University)
Ganchev, Georgi (Institute of Mathematics and Informatics Bulgarian Academy of Sciences)
Milousheva, Velichka (Institute of Mathematics and Informatics Bulgarian Academy of Sciences)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1077-1100 More about this Journal
Abstract
We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.
Keywords
Lorentz surface; fundamental existence and uniqueness theorem; parallel normalized mean curvature vector; canonical parameters;
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1 B.-Y. Chen, Pseudo-Riemannian geometry, ${\delta}$-invariants and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
2 B.-Y. Chen, Classification of minimal Lorentz surfaces in indefinite space forms with arbitrary codimension and arbitrary index, Publ. Math. Debrecen 78 (2011), no. 2, 485-503.   DOI
3 B.-Y. Chen and F. Dillen, Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms, J. Math. Phys., 48 (2007), no. 1, 013509, 23 pp.; Erratum, J. Math. Phys. 49 (2008), no. 5, 059901, 1 p.
4 B.-Y. Chen, F. Dillen, and J. Van der Veken, Complete classification of parallel Lorentzian surfaces in Lorentzian complex space norms, Internat. J. Math. 21 (2010), no. 5, 665-686.   DOI
5 B.-Y. Chen and O. Garay, Classification of quasi-minimal surfaces with parallel mean curvature vector in pseudo-Euclidean 4-space $\mathbb{E}^4_2$, Result. Math. 55 (2009), no. 1-2, 23-38.   DOI
6 B.-Y. Chen and I. Mihai, Classification of quasi-minimal slant surfaces in Lorentzian complex space forms, Acta Math. Hungar. 122 (2009), no. 4, 307-328.   DOI
7 B.-Y. Chen and J. Van der Veken, Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms, Tohoku Math. J. 61 (2009), no. 1, 1-40.   DOI
8 B.-Y. Chen and D. Yang, Addendum to "Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its application to biharmonic surfaces", J. Math. Anal. Appl. 361 (2010), no. 1, 280-282.   DOI
9 Y. Fu and Z.-H. Hou, Classification of Lorentzian surfaces with parallel mean curvature vector in pseudo-Euclidean spaces, J. Math. Anal. Appl. 371 (2010), no. 1, 25-40.   DOI
10 G. Ganchev and V. Milousheva, An invariant theory of spacelike surfaces in the fourdimensional Minkowski space, Mediterr. J. Math. 9 (2012), no. 2, 267-294.   DOI
11 G. Ganchev and V. Milousheva, An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space, J. Math. Phys. 53 (2012), no. 3, 033705, 15 pp.   DOI
12 G. Ganchev and V. Milousheva, Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space, Cent. Eur. J. Math. 12 (2014), no. 10, 1586-1601.
13 S. Haesen and M. Ortega, Screw invariant marginally trapped surfaces in Minkowski 4-space, J. Math. Anal. Appl. 355 (2009), no. 2, 639-648.   DOI
14 E. Lane, Projective Differential Geometry of Curves and Surfaces, University of Chicago Press, Chicago, 1932.
15 J. Little, On singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl. IV Ser. 83 (1969), 261-335.   DOI
16 R. Rosca, On null hypersurfaces of a Lorentzian manifold, Tensor (N.S.) 23 (1972), 66-74.
17 S. Shu, Space-like submanifolds with parallel normalized mean curvature vector field in de Sitter space, J. Math. Phys. Anal. Geom. 7 (2011), no. 4, 352-369.
18 R. Walter, Uber zweidimensionale parabolische Flachen im vierdimensionalen affinen Raum. I: Allgemeine Flachentheorie, J. Reine Angew. Math. 227 (1967), 178-208.
19 S.-T. Yau, Submanifolds with constant mean curvature, Amer. J. Math. 96 (1974), 346-366.   DOI
20 C. Burstin and W. Mayer, Uber affine Geometrie XLI: Die Geometrie zweifach ausgedehnter Mannigfaltigkeiten $F_2$ im affinen $R_4$, Math. Z. 26 (1927), no. 1, 373-407.   DOI
21 B.-Y. Chen, Geometry of submanifolds, Marcel Dekker, Inc., New York, 1973.
22 B.-Y. Chen, Surfaces with parallel normalized mean curvature vector, Monatsh. Math. 90 (1980), no. 3, 185-194.   DOI
23 B.-Y. Chen, Classification of spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces with arbitrary codimension, J. Math. Phys. 50 (2009), 043503.   DOI
24 B.-Y. Chen, Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its application to biharmonic surfaces, J. Math. Anal. Appl. 340 (2008), no. 2, 861-875.   DOI
25 B.-Y. Chen, Classification of marginally trapped surfaces of constant curvature in Lorentzian complex plane, Hokkaido Math. J. 38 (2009), no. 2, 361-408.   DOI
26 B.-Y. Chen, Black holes, marginally trapped surfaces and quasi-minimal surfaces, Tamkang J. Math. 40 (2009), no. 4, 313-341.
27 B.-Y. Chen, Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms, Cent. Eur. J. Math. 7 (2009), 400-428.   DOI
28 B.-Y. Chen, Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space, Cent. Eur. J. Math. 8 (2010), no. 4, 706-734.   DOI
29 B.-Y. Chen, Complete classification of parallel Lorentz surfaces in four-dimensional neutral pseudosphere, J. Math. Phys. 51 (2010), no. 8, 083518, 22 pp.   DOI
30 B.-Y. Chen, Complete explicit classification of parallel Lorentz surfaces in arbitrary pseudo-Euclidean spaces, J. Geom. Phys. 60 (2010), no. 10, 1333-1351.   DOI
31 B.-Y. Chen, Complete classification of Lorentz surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean space, Kyushu J. Math. 64 (2010), no. 2, 261-279.   DOI
32 B.-Y. Chen, Submanifolds with parallel mean curvature vector in Riemannian and indefinite space forms, Arab J. Math. Sci. 16 (2010), no. 1, 1-46.