Isometries of 2-Dimensional Riemannian Manifolds into Themselves.
Isometries of 2-Dimensional Riemannian Manifolds into Themselves.
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DOI：
10.1073/pnas.22.5.297
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发表时间：
1936-05
影响因子：
11.1
通讯作者：
S. Myers
中科院分区：
文献类型：
--
作者：
S. Myers
1.引言--这篇摘要包含了二维黎曼流形到它们自身的等距关系的研究结果。这种研究实质上是对小2中黎曼空间运动群的研究和对二维流形到自身的拓扑变换的研究的结合。本课题与共形映射理论和一般度量空间到自身的等距理论有密切的联系。这里我们证明了紧致的二维黎曼流形的等距群是连续的(即局部欧式的)或有限的；用W.Rinow提出的方法研究连续群，用作者在前文中研究的“最小点轨迹”研究周期等距线。7.关于n维黎曼Mani折叠的等距的一般概念-设M是具有C4类直线元素的5类正则n维流形。M到自身的等距被定义为保长度同胚。通过将两点之间的距离定义为连接两点的可直圆弧的长度的最大下界，我们将M定义为度量空间。定理1.n维黎曼流形M的保距同胚与保长同态射构成一个可度量化群G。如果M是紧的，则两个等距线T之间的距离T和T2可以定义为M上P的最大距离T，(P)‘T2(P)，该群是度量群。
1. Introduction.-This abstract contains the results of a study of the isometries of 2-dimensional Riemannian manifolds' into themselves. Such a study is essentially a combination of a study of groups of motions of Riemannian spaces in the small2 with a study of topological transformations of 2-dimensional manifolds into themselves.'The subject comes into close contact with the theory of conformal mapping'and the theory of isometries of general metric spaces into themselves.'We show here that the group of isometries of a compact 2-dimensional Riemannian manifold is either continuous (ie, locally euclidean) or finite; the continuous groups are studied by means of methods initiated by W. Rinow, 6 while a method is devised for the study of periodic isometries by means of the" minimum point locus" studied by the author in previous papers. 7 2. GeneralNotions about Isometries of n-Dimensional Riemannian Mani folds.-Let M be a regular n-dimensional manifoldof class 5 with a line element of class C4. An isometry of M into itself is defined to be a length-preserving homeomorphism. We make M into a metric space by defining the distance between two points to be the greatest lower bound of the lengths of rectifiable arcs joining thetwo points. THEOREM 1. The distance preserving homeomorphisms of an n-dimensional Riemannian manifold M are identical with the length-preserving homeo-morphisms.The isometries of M into itself form a metrizable group G. If M is com-pact, the distance between two isometries T, and T2 can be defined to be the maximum distance T,(P)'T2 (P) for P on M, and the group is a metric8 group.