Classification Of Non-developable Ruled Surfaces In The Minkowski 3-space

Ruled surface is a well-known geometric object. It has been an important research topic to study the geometric properties of ruled surface in the three-dimensional Minkowski space. With the introduction of the second Gaussian curvature of surface, there have been further in-depth studies of ruled surface. In this paper, we mainly investigate non-developable ruled surfaces in which second Gaussian curvature, mean curvature and Gaussian curvature satisfy some prescribed conditions and give them classifications. This paper is arranged as follows:In chapter 1, we review the historical development and some important consequences.In chapter 2, we introduce the definitions and properties of inner product and outer product of three-dimensional Minkowski space and the basic knowledge of curves, planes and surfaces in the three-dimensional Minkowski space.In chapter 3, we introduce the definitions of Gauss curvature(K), mean curvature(H) and second Gauss curvature(H_II) of ruled surface in the three-dimensional Minkowski space and give some examples and graphics of non-developable ruled surface.In chapter 4,we know that Young Ho Kim and Dae Won Yoon had discussed the classifications of ruled surface respectively when K_II, if, K pairwise satisfy some linear equations such that aK_II+bH = const, 2a - b≠0, aH+bK = const, a≠0oraK_II+bK = const, a≠0 in the three-dimensional Minkowski space in reference [15]. On the base of reference [15], we investigate the classifications of ruled surface respectively when K_II, H, K pairwise satisfy some nonlinear equations such thata₁K_II + a₂H + a₁K_II² + a₂₂H² + a₁₂K_IIH = c, a₁H + a₂K + a₁₁H² + a₂₂K² + a₁₂HK = c,a₁K_II + a₂K + a₁₁K_II² + a₂₂K² + a₁₂K_IIK = c,in which a_i, a_ij(i,j = 1,2, i≤j), c are constants and a₁≠0. Meanwhile, we give some conclusions in order to enrich the theory of ruled surface in the three-dimensional Minkowski space.