【作者】 李旭；

【导师】 陈亮；

【作者基本信息】 东北师范大学 ， 基础数学， 2021， 硕士

【摘要】 众所周知,三维双曲空间是闵科夫斯基空间中的伪球空间之一,双曲几何（Hyperbolic Geometry）和极限圆几何（Horospherical Geometry）都是三维双曲空间中的重要几何,而本文前半部分主要研究了极限圆几何,即利用曲线的伏雷内型公式和达布向量场构造了两种沿着给定曲面上正则曲线的极限圆曲面和极限圆平坦曲面,分别为切极限圆平坦曲面和法极限圆平坦曲面,这两种曲面在给定曲线的任意点处分别切于和垂直于给定的曲面,最后讨论了这些极限圆平坦曲面出现尖棱、燕尾、尖喙以及交叉帽型奇点需满足的条件.极限圆曲面和极限圆平坦曲面的关系类似于欧氏空间中的直纹面和可展曲面,可展曲面是高斯映射退化为一点或者一条曲线的直纹面,而曲面的勒让德对偶扮演着类似于高斯映射的角色,若曲面的勒让德对偶在任意点处均奇异,那么该曲面具有某种平坦性,其中若曲面的Δ₂-对偶在任意点处均奇异,则该曲面具有极限圆平坦性.沿着曲面上曲线的极限圆平坦曲面可看作是原曲面在曲线某点处的近似平坦.除了双曲几何和极限圆几何,还有一种介于二者之间的几何,称为斜几何.于是本文接着利用斜几何的理论构造了沿着曲面上一条正则曲线的勒让德对偶曲面,自然地,利用曲面勒让德对偶的退化性定义了勒让德对偶曲面的斜平坦性,勒让德对偶曲面和斜平坦曲面的关系类似于极限圆曲面和极限圆平坦曲面.除此之外,受到Saji的沿着尖棱构造极限圆平坦曲面的启发,本文还构造了两种沿着尖棱的勒让德对偶曲面和斜平坦曲面,分别称为切斜平坦曲面和法斜平坦曲面,类似的发现,这两种斜平坦曲面在尖棱处切于或垂直于尖棱.最后讨论了这些斜平坦曲面的奇点类型.同理,沿着曲面上曲线的这些斜平坦曲面可看作原曲面在曲线某点处的斜平坦近似.最后,当给定曲面上的曲线是特殊曲线时,例如是曲率线时,分别构造了极限圆平坦曲面和斜平坦曲面,并且研究了这些近似（斜）平坦曲面具有的微分几何性质,最后发现这些曲面出现了纯的frontal奇点.本文以构造给定曲面上曲线的近似（斜）平坦曲面,并且研究其微分几何性质为主要思路,不仅研究了曲面上一般正则曲线的近似（斜）平坦曲面,而且也研究了特殊曲线,例如沿着尖棱或是曲率线的近似（斜）平坦曲面.构造这些近似（斜）平坦曲面的意义就在于,直接研究曲面或曲线本身的几何性质较为困难时,就可先构造沿着曲线的近似（斜）平坦曲面,而这些近似（斜）平坦曲面可看作曲线上某点处在原曲面上的近似曲面,这样就可以利用曲面在曲线某点处的平坦逼近的特殊性质,研究曲面或者曲线的几何性质.不仅简化了研究过程,而且可反观出原曲面或者曲线所具有的几何性质.更多还原

【Abstract】 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 itsΔ₂-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.更多还原