The ?-flat Surfaces In The Hyperbolic 3-space

As is known to all,the hyperbolic 3-space is one of pseudo-spheres in Minkowski space.Both hyperbolic geometry and horospherical geometry are important geometry in hyperbol-ic 3-space.Therefore,in the first half of this paper,we mainly studied the horospherical geometry.By using a moving frame and Darbox vector field along the curve on the surface,we have constructed two kinds of horocyclic surfaces and horo-flat surfaces.These two sur-faces are tangent to and normal to the given surface at any point of the curve respectively.Finally,the condition for singularities are discussed.Not only cuspidal edges and swallow-tails appear,but also cuspidal lips and cuspidal cross caps as well.The horocyclic surfaces and horo-flat surfaces are analogous notion to ruled surfaces and developable surfaces in the Euclidean space.It's well-known that a surface is developable if the image of its Gauss map is a point or a curve.Furthermore,the Legendrian dual of a surface plays a similar role to the Gauss map of the surfaces.So the horo-flatness of surfaces can be defined by the degeneracy of its Legendrian dual.In particular,we can define the horo-flatness by using the degeneracy of its?₂-dual.Such horo-flat surfaces along the curve on the surface can be considered as flat approximations of the given surface at some point on the curve.In addition to hyperbolic geometry and horospherical geometry,there is another geom-etry in between,called slant geometry.Taking into account these theory,we constructed two kinds of Legendrian dual surfaces along a regular curve on the surface.Naturally,we defined the slant-flatness by using the degeneracy of its Legendrian dual.The relationship between Legendrian dual surfaces and slant-flat surfaces is also similar to that between horocyclic surfaces and horo-flat surfaces.What's more,inspired by Saji's construction of horo-flat surfaces along the cuspidal edge,we can also investigate two kinds of Legendrian dual surfaces and slant-flat surfaces along the cuspidal edge,which are tangent and normal slant-flat surfaces.Similarly,these two surfaces are found to be tangent to or normal to the cuspidal edge.At last,the singularity types are discussed.In a same way,these slant-flat surfaces can be considered as slant flat approximations of the given surface at the point on the curve.Finally,if the curve on a given surface is a special curve,such as lines of curvature,horo-flat surfaces and slant-flat surfaces are constructed respectively,and we also studied the differential geometric of these surfaces.And it turns out that pure frontal singular points appear.In this paper,the main idea is to construct(slant)flat approximations along the curve on the given surface and study its differential geometric properties,so we studied not only the(slant)flat approximations along the regular curve on the surface,but also the special curve,such as the cuspidal edge and lines of curvature.The significance of constructing these surfaces is that when it is difficult to study the surface or the curve,we can construct(slant)flat surfaces along the curve on the surface,and these surfaces can be viewed as ap-proximations of the surface at some point on the curve,and then we can turn to investigate the properties of these(slant)flat approximations surfaces.We investigate the geometrical properties of a surface or a curve in terms of the special properties of its flat approximations.In this way,the research process is not only simplified,but also the geometric properties of the original surface or curve can be reflected.