J. Korean Math. Soc. 2016; 53(5): 1077-1100
Printed September 1, 2016
https://doi.org/10.4134/JKMS.j150381
Copyright © The Korean Mathematical Society.
Yana Aleksieva, Georgi Ganchev, and Velichka Milousheva
5 James Bourchier blvd., Acad. G. Bonchev Str. bl. 8, 175 Suhodolska Str.
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
MSC numbers: Primary 53B30; Secondary 53A35, 53B25
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