On a theorem of Alekseevskii concerning conformal transformations
On a theorem of Alekseevskii concerning conformal transformations
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论阿列克谢耶夫斯基关于共形变换的定理
DOI：
10.2969/jmsj/02820278
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发表时间：
1976
影响因子：
0.7
通讯作者：
Yashiro Yoshimatsu
中科院分区：
文献类型：
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作者：
Yashiro Yoshimatsu
本文的目的是给出Alekseevskii定理的一个直接证明，该定理断言在黎曼流形的某一点周围存在一种特殊的邻域。设$M$是一个维数为$m\geqq 3$的黎曼流形。以下被称为Lichnerowicz猜想：如果$M$的共形变换的最大连通群$C_{0}（M）$是本质的（术语的含义见\S 1），则$M$要么共形于$a$欧几里德球面$S^{m}$，要么共形于$a$欧几里德空间$E^{m}$。Lelong-Ferrand [2]和Obata [3]在M$是紧的情况下肯定地回答了这个猜想，其他人在一些附加条件下也肯定地回答了这个猜想。[4]）。最近Alekseevskii [1]试图确保最一般情况下的猜想。本文所要建立的定理在文[1]中是以一种较弱的形式叙述的，其证明似乎是不完全的。我们将以一种与Alekseevskii的稍有不同的方式证明我们的定理如何应用于Lichnerowicz猜想的证明，在假设$M$允许$C_{0}（M）$的本质单参数子群的一般情况下。作者谨向T教授表示衷心的感谢。Ochiai感谢他的善意建议和感谢。
The purpose of this note is to give a direct proof to a theorem of Alekseevskii which asserts existence of a special kind of neighborhoods around a certain point of a Riemannian manifold. Let $M$ be a Riemannian manifold of dimension $m\geqq 3$ . The following is known as Lichnerowicz’s conjecture: If the largest connected group $C_{0}(M)$ of conformal transformations of $M$ is essential (See \S 1 for the meaning of terminology), then $M$ is conformal either to $a$ Euclidian sphere $S^{m}$ or to $a$ Euclidian space $E^{m}$. This conjecture is affirmatively answered by Lelong-Ferrand [2] and by Obata [3] in the case when $M$ is compact, and also by others under some additional conditions (cf. [4]). Recently Alekseevskii [1] tried to assure the conjecture for the most general case. The theorem we shall establish in this paper is stated in a slightly weaker form in the paper [1] with a proof which seems incomplete. We shall show, in a way slightly different from Alekseevskii’s, how our theorem is applied to a proof of Lichnerowicz’s conjecture for the general case under the assumption that $M$ admits an essential one-parameter subgroup of $C_{0}(M)$ . The author wishes to express his hearty thanks to Professor T. Ochiai for his kind advices and encouragements.