On submanifolds of submanifolds of a Riemannian manifold
On submanifolds of submanifolds of a Riemannian manifold
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DOI：
10.2969/jmsj/02330548
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发表时间：
1971-07
影响因子：
0.7
通讯作者：
Bang‐Yen Chen;K. Yano
中科院分区：
文献类型：
--
作者：
Bang‐Yen Chen;K. Yano
Takahashi [1] proved that (i) in order for a submanifold $M^{n}$ of an mdimensional euclidean space $E^{m}$ to be a minimal submanifold it is necessary and sufficient that the radius vector $X$ satisfies $\Delta X=0$, where $\Delta$ denotes the Laplacian in the submanifold $M$“, $i$ . $e$ . all the (natural) coordinate functions are harmonic, and (ii) in order that a submanifold of a hypersphere with radius $r$ of a euclidean space is minimal, it is necessary and sufficient thatr the radius vector $X$ satisfies $\Delta X=(-n/r^{2})X$, where $n$ denotes the dimension of the submanifold. The main purpose of the present paper is to study, a submanifold $M^{n}$ of a submanifold $M^{m}$ of a Riemannian manifold $M^{l}$ being given, the conditions that $M^{n}$ is minimal in $M^{m}$ or that $M^{n}$ is minimal in $M^{\iota}$ . and to obtain a theorem which generalizes two results above of Takahashi.
