Geometric Meaning Of The Second Fundamental Form For Surfaces In Minkowski 3-Space

In Euclidean space,the degree of curvature of a surface is described by the second fundamental form of the surface,the specific meaning of which is that the second fundamental form Ⅱ of the surface is approximately equal to twice the directional distance between the surface and the tangent plane.With the development of indefinite metric geometry,many geometrical problems in Euclidean space are generalized to Minkowski space.Only the study of the geometric meaning of the second fundamental form of a surface in Minkowski space is of great importance.Due to the uncertainty of the measurement in Minkowski space,the vectors can be divided into three categories:spacelike,timelike and lightlike vectors.According to the classification of the vectors in the space,the curves and the surfaces are also classified.In this paper,accordingly the classification of curves,the classification of vectors and the moving frame method,the geometrical meanings of the second fundamental form of different surfaces in three-dimensional Minkowski space are discussed.The second chapter introduces the basic knowledge of space,including the classification of curves and surfaces and the activities of the frame,etc.The third chapter and the fourth chapter study the second fundamental form of nondegenerate timelike and spacelike surface and its geometric significance.In chapter 5,we study the degenerated surface,that is,the geometrical meanings of the second fundamental form of the lightlike surface.At the same time,we conclude that the second fundamental form of the non-degenerate surface in the three-dimensional Minkowski space is approximately twice the directional distance between the surface and the tangent plane,While the second fundamental form of a lightlike surface approximately equals twice the directional distance from the surface to the normal,or twice the distance between the surface and the plane normal to the tangent vector.