Certain conditions for a Riemannian manifold to be isometric with a sphere
Certain conditions for a Riemannian manifold to be isometric with a sphere
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DOI：
10.2969/jmsj/01430333
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发表时间：
1962-07
影响因子：
0.7
通讯作者：
M. Obata
中科院分区：
文献类型：
--
作者：
M. Obata
介绍。在本文中，黎曼流形始终是指维度为 $n(\geqq 2)$ 且具有正定 $C^{\infty}$ -黎曼度量的连通 $C^{\infty}-$ 流形。如果黎曼流形的变换保留黎曼度量定义的角度，则该变换被认为是保形的。显然，对于一个黎曼度量的共形变换对于与原始度量共形相关的度量也是共形的。众所周知[5]，如果流形是紧致的并且维数大于二，则其上的每个黎曼度量都可以共形变形为具有恒定标量曲率的流形。因此，就保角变换而言，我们可以假设上述情况下标量曲率恒定。然后，通过变换保留恒定的标量曲率。这个性质似乎有助于共形变换的研究。事实上，两个非正标量曲率的紧致黎曼流形之间的共形变换（不一定是常数）是等距的，当且仅当它将一个标量曲率带入另一个标量曲率时[4]。因此，仅当标量曲率为正时，恒定标量曲率的紧黎曼流形才允许共角变换，该变换不是等距的。因此，就保形变换而言，我们可以将我们的考虑限制在正常数标量曲率的黎曼流形上，前提是流形是紧的并且维度为 $>2$ 。此外，如果恒定标量曲率 $k$ 的黎曼流形允许无穷小共角变换 $u$，其中 $C\vee u\iota\cdot g=2\phi g$，其中 $g$ 是黎曼度量，$c_{u}$ 是对应于 $u$ 和 $\phi$ 函数的李导数的运算，则 $\phi$ 满足方程 $\Delta\phi=nk\phi$ ，其中$k=$
Introduction. In this paper, by a Riemannian manifold we always mean a connected $C^{\infty}-$ manifold of dimension $n(\geqq 2)$ with a positive definite $C^{\infty}$ -Riemannian metric. A transformation of a Riemannian manifold is said to be conformal if it preserves the angle defined by the Riemannian metric. Evidently a conformal transformation with respect to one Riemannian metric is also conformal with respect to one conformally related to the original one. It is known [5] that if a manifold is compact and of dimension greater than two, every Riemannian metric on it can be conformally deformed into one with constant scalar curvature. Thus as far as a conformal transformation is concerned, we may assume the constancy of the scalar curvature in the above case. Then the scalar curvature, which is constant, is preserved by the transformation. This property seems to help the study of conformal transformations. Indeed, a conformal transformation between two compact Riemannian manifolds of nonpositive scalar curvature, not necessarily constant, is isometric if and only if it carries one scalar curvature into another [4]. Thus a compact Riemannian manifold of constant scalar curvature admits a conformal transformation, which is not isometric, only if the scalar curvature is positive. Therefore we may restrict our consideration to Riemannian manifolds of positive constant scalar curvature as far as a conformal transformation is concerned, provided that the manifold is compact and of dimension $>2$ . Furthermore, if a Riemannian manifold of constant scalar curvature $k$ admits an infinitesimal conformal transformation $u$ with $C\vee u\iota\cdot g=2\phi g$, where $g$ is the Riemannian metric, $c_{u}$ the operation of Lie derivatives corresponding to $u$ and $\phi$ a function, then $\phi$ satisfies the equation $\Delta\phi=nk\phi$ , where $k=$