On Lorentz GCR Surfaces in Minkowski 3-Space

Abstract

Abstract. A generalized constant ratio surface (GCR surface) is defined by the property that the tangential component of the position vector is a principal direction at each point on the surface, see [8] for details. In this paper, by solving some differential equations, a complete classification of Lorentz GCR surfaces in the three-dimensional Minkowski space is presented. Moreover, it turns out that a flat Lorentz GCR surface is an open part of a cylinder, apart from a plane and a CMC Lorentz GCR surface is a surface of revolution.

On Generalized Constant Ratio Surfaces of Higher Codimension

“…By a further computation considering (12) and the last two equations, we obtain the following lemma, where we put S e β = S β and h β ij = h(e i , e j ), e β .…”

“…4). Recently in [12,18] the completed classification of GCR surfaces in Minkowski 3-spaces has been obtained. Also in [11] and [12,18], all flat and constant mean curvature GCR surfaces, respectively, in E 3 and E 3 1 were classified.…”

“…Recently in [12,18] the completed classification of GCR surfaces in Minkowski 3-spaces has been obtained. Also in [11] and [12,18], all flat and constant mean curvature GCR surfaces, respectively, in E 3 and E 3 1 were classified. Before we proceed to the other definition that we would like to give, we want to mention about class A immersions into the product spaces Q n c × R, where Q n c denotes the Riemannian space form with dimension n and sectional curvatures c = ±1.…”

A Short Survey on Surfaces Endowed with a Canonical Principal Direction

“…Most recently, Lorentzian GCR surfaces in the 3-dimensional Minkowski space investigated by Fu and Yang in [11]. …”

“…On the other hand, in [8,9,11,23] authors study generalized constant ratio surfaces. A hypersurface M in a semi-Euclidean space E n+1 t is said to be a generalized constant ratio surfaces if the tangential component of its position vector is a principal direction of M .…”