A new curvature theory for surfaces in a euclidean 4-space
A new curvature theory for surfaces in a euclidean 4-space
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欧几里得 4 空间曲面的新曲率理论
DOI：
10.1007/bf02564298
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发表时间：
1952
期刊：
影响因子：
--
通讯作者：
Y. Wong
中科院分区：
文献类型：
--
作者：
Y. Wong
在欧几里得4空间R4中曲面局部性质的研究中，我们的注意力迄今集中在Kommerell圆锥（Kommerell[7]，第553页）和曲率椭圆（Schouten-Struik[7]，第104—111页）。设~，~’为曲面(a)在点a处的切面和法平面，则(a)在点a处的Kommerell二次曲线(K)是~’与(a)连续于~’的法平面交点K的轨迹。曲率椭圆(G)，也在~'中，得到如下：设J为(A)在A处的任意切单位向量，(C)为(A)上与J相切的任意曲线；那么(C)在A处的曲率向量在~'中的分量，相对于R4，只依赖于J （~ r定理）；当J在~中取所有方向时，这个分量的终点轨迹是曲率椭圆(G)。二次曲线(K)和(G)相对于~'中的单位圆是极倒数。解析地说，将(G)引入到R4中曲面的研究中是很自然的，因为(G)与(A)的gaas - codazzi - ricci方程所依赖的(A)的两种基本形式密切相关。然而，从几何上讲，引入(K)更为自然。的事实,第一个定义曲线的曲率的变化率连续两个切线的夹角,这很奇怪,没有系统的研究已经取得了相应的所扮演的角色之间的两个角(参看1.4 w)一对连续切平面的表面在R。据作者意识到,唯一已知的结果,这两个角是共轭方向起到直接或间接的作用,Kwietniewski-Kommerell-Eisenhart定理（w1.3），以及riemanan n空间中m曲面上的Struik[12]的“主方向”，对于R4中的曲面，它与w1.5中定义的函数~的主方向相同。本文的目的是提出R4中曲面的曲率理论，该曲率理论是基于曲面的两个连续切平面之间的夹角
In the study of local properties of surfaces in the Euclidean 4-space R4, our attention has so far centred on the Kommerell conic (Kommerell [7], p. 553) and the curvature ellipse (Schouten-Struik [11], pp. 104--111). Let~,~'be the tangent and normal planes of a surface (A) at the point A. Then the Kommerell conic (K) of (A) at A is the locus of the point K of intersection of~'by the normal planes of (A) consecutive to~'. The curvature ellipse (G), also lying in~', is obtained as follows. Let J be any tangent unit vector of (A) at A, and (C) any curve on (A) tangent to J at A; then the component in~'of the curvature vector of (C) at A, with respect to R4, depends only on J (~ r Theorem); the locus of the end point of this component as J takes on all the directions in~ is the curvature ellipse (G). The conics (K) and (G) are polar reciprocal of each other with respect to the unit circle in~'. Analytically, the introduction of (G) into the study of surfaces in R4 is quite natural because (G) is tied up closely with the two fundamental forms of (A) on which the Gauss-Codazzi-Ricci equations of (A) depend. Geometrically, however, the introduction of (K) is more natural. In view of the fact that the first curvature of a curve is defined to be the rate of change of the angle between two consecutive tangent lines, it is rather surprising that no systematic study has been made of the corresponding role played by the two angles (cf. w 1.4) between a pair of consecutive tangent planes of a surface in R a. As far as the author is aware, the only known results in which these two angles play a direct or indirect part are the conjugate directions, the Kwietniewski-Kommerell-Eisenhart theorem (w 1.3), and the" principal directions" of Struik [12] on an m-surface in a Riemannian n-space, which for a surface in R4 are identical with the principal directions of the function~ defined later in w 1.5. The purpose of this paper is to present a curvature theory for surfaces in R4 based on the two angles between consecutive tangent planes of the