Minimal immersions of Riemannian manifolds
Minimal immersions of Riemannian manifolds
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DOI：
10.2969/jmsj/01840380
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发表时间：
1966-10
影响因子：
0.7
通讯作者：
Tsunero Takahashi
中科院分区：
文献类型：
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作者：
Tsunero Takahashi
黎曼流形$M在另一个流形中的等距浸入称为极小，如果它的每个平均曲率为零。本文将讨论常曲率空间中黎曼流形的极小浸入问题。在S 1中，我们总结了众所周知的关于浸入的符号和公式，并给出了黎曼流形在常曲率空间中最小浸入的一个判据(定理1)。在S 2中，我们将讨论黎曼流形在$(m+k)$维欧氏空间$R^{m+k}$中的浸入问题。如果$M$乘以$x$的像$x(M)$包含在$R^{m+k}$中的$(m+k-1)$维球面$S^{m+k-1}$中，我们称之为浸没$x$实现了球面中的浸没。由于浸没函数$x$可以看作是$M$上的向量值函数，因此我们可以将Laplace-Beltrami算子$\Delta$应用于$x$。定理2断言浸没$x$是极小的当且仅当$\Delta x=0$，定理3断言浸入$x$实现球面上的最小浸没当且仅当$\Delta x=\lambda x$对于某个常数$\lambda\neq 0$且球体的半径完全由$\lambda$决定。定理2是由J.Eells和J.H.Sampson[1]得到的。在S 3中，我们将给出一个黎曼流形的例子，它允许在满足$\Delta x=\lambda x$的欧氏空间中有一个浸入，并证明了具有不可约线性迷向群的紧致齐次黎曼流形允许在球面上有一个极小浸入。这个例子是由T.Nagano的一项工作[2]激发的。作者感谢T.Nagano教授和M.Obata教授在这项研究中提出了许多宝贵的建议。
An isometric immersion $M\rightarrow M^{\prime}$ of a Riemannian manifold $M$ in anothet manifold $M^{\prime}$ is called to be minimal, if each of its mean curvatures vanishes. In this paper we shall deal with minimal immersions of Riemannian manifolds in a space of constant curvature. In \S 1 we shall summarize notations and formulas concerning immersions which are all well-known, and give a criterion for a Riemannian manifold to be immersed minimally in a space of constant curvature (Theorem 1). In \S 2 we shall deal with an immersion $x:M\rightarrow R^{m+k}$ of a Riemannian mmanifold in an $(m+k)$-dimensional Euclidean space $R^{m+k}$ . If the image $x(M)$ of $M$ by $x$ is contained in an $(m+k-1)$ -dimensional sphere $S^{m+k-1}$ in $R^{m+k}$ , we shall call that the immersion $x$ realizes an immersion in a sphere. Since the immersion $x$ can be considered as a vector valued function on $M$, we can apply Laplace-Beltrami operator $\Delta$ to $x$ . Theorem 2 asserts that the immersion $x$ is minimal if and only if $\Delta x=0$ , and Theorem 3 asserts that the immersion $x$ realizes a minimal immersion in a sphere if and only if $\Delta x=\lambda x$ for some constant $\lambda\neq 0$ and the radius of the sphere is completely determined by $\lambda$ . Theorem 2 has been obtained by J. Eells and J. H. Sampson [1]. In \S 3 we shall give an example of a Riemannian manifold which admits an immersion $\chi$ in a Euclidean space satisfying $\Delta x=\lambda x$ , and prove that the compact homogeneous Riemannian manifold with irreducible linear isotropy group admits a minimal immersion in a sphere. This example is motivated by a work of T. Nagano [2]. The author is grateful to Professors T. Nagano and M. Obata for their many valuable suggestions in this research.