Function-theoretic characterization of Einstein spaces and harmonic spaces
Function-theoretic characterization of Einstein spaces and harmonic spaces
复制标题
爱因斯坦空间和调和空间的函数论表征
DOI：
10.1090/s0002-9947-1961-0131839-8
复制
发表时间：
1961
影响因子：
1.3
通讯作者：
A. Friedman
中科院分区：
文献类型：
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作者：
A. Friedman
导言。考虑以下一般问题：设P是一个函数论性质，它在(局部或整体)欧几里德空间E的点上成立，并且可以形式地表示为黎曼空间Rn的点。找出一个语句Q，它是R的一个内在(几何)性质，使得P蕴含Q，Q蕴含P。相反，给定Q，人们可以寻找P。在本文中，我们解决上述类型的特殊问题。我们把R(总是假设它有一个正定度量)是一个爱因斯坦空间，一个具有特定基本解的调和空间(将在？3中定义)和一个常曲率空间作为Q的陈述。则P代表关于某些方程的解的平均值的各种表述。用M(u，x？，R)表示以x为中心的测地球面上u的平均值？和半径R，我们得到如下结果：爱因斯坦空间(在？1中)被刻画为
Introduction. Consider the following general problem: Let P be a function-theoretic property which holds at the points of a (local or global) Euclidean space E. and which can be stated, formally, for points of a Riemannian space Rn. Find a statement Q which is an intrinsic (geometric) property of R. such that P implies Q and Q implies P. Conversely, given Q, one may look for P. In this paper we solve particular problems of the above type. We take for Q the statements that R. (which is always assumed to have a positive definite metric) is an Einstein space, an harmonic space (to be defined in ?3) with a particular fundamental solution, and a space with constant curvature. P then stands for various statements about the mean value of solutions of certain equations. Denoting by M(u, x?, R) the mean value of u on the geodesic sphere with center x? and radius R we obtain the following results: An Einstein space is characterized (in ?1) by