Year: 1989
Type: article
Abstract: Let $M^{n}$ be an n-dimensional Riemannian manifold which is minimally immersed in a unit sphere $S^{n+p}(1)$ of dimension $n+p$ .If $M^{n}$ is compact, then many authors studied them and obtained many beautiful results (for examples [1],[3], [4], [5] and [6]).In this paper, we make use of Yau's maximum principle to extend these results to complete manifolds with Ricci curvature bounded from below. Preliminaries.Let $M^{n}$ be an n-dimensional Riemannian manifold which is minimally immersed in a unit sphere $S^{n+p}(1)$ of dimension $n+p$ .Then the second funda- mental form $h$ of the immersion is given by $h(X, Y)=\tilde{\nabla}_{X}Y-\nabla_{X}Y$ and it satisfies $h(X, Y)=h(Y, X)$ , where $\tilde{\nabla}$ and $\nabla$ denote the covariant differentiation on $S^{n+p}(1)$ and $M^{n}$ respectively, $X$ and $Y$ are vector fields on $M^{n}$ .We choose a local field of orthonormal frames $e_{1},$ $\cdots$ , $e_{n},$ $\cdots,$ $e_{n+p}$ in $S^{n+p}(1)$ such that, restricted to $M^{n}$ , the vector $e_{1},$ $\cdots$ , $e_{n}$ are tangent to $M^{n}$ .We use the following conven-
Source: Tsukuba Journal of Mathematics
Authors: Qing-ming Cheng, Yuen-da Wang
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1584331691