摘要
C~∞(R^2)表示欧氏平面R^2中全体简单、光滑、闭曲线,它构成以C_(2x)~∞为模空间的Frechet流形。本文则是在C~∞(R^2)中定义了一种自然度量,使其度量拓扑与流形拓扑等价同时得到了C~∞(R^2)中保持这种度量的联络,从而为进一步研究C~∞(R^2)的几何性质奠定基础。
C~∞(R^2) represents all of the simple closed curves in Euclidean plane R^2. It constitutes a Frechet manifold with modul space C_2(?)~∞. In this paper, a natural measure is defined in C~∞(R^2), such that the topology of measure is equivalent to the topology of manifold, and that C~∞(R^2) is obtained to preserve the connection of the measure. Thus the basis for the further study of more geometric properties on C~∞(R^2) is given,
