## Abstract

In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere ${S}^{n+p}\left(1\right)$. First of all, we proved that, for an $n$-dimensional $(n\ge 2)$ submanifold $\mathbf{x}:M\mapsto {S}^{n+p}\left(1\right)$ without umbilical points and with vanishing Möbius form $\mathrm{\Phi}$, if $(n-2)\left|\right|\tilde{\mathbf{A}}\left|\right|\le \sqrt{\frac{n-1}{n}}\left\{nR-\frac{1}{n}\left[\right(n-1)\left(2-\frac{1}{p}\right)-1]\right\}$ is satisfied, then, $\mathbf{x}$ is Möbius equivalent to an open part of either the Riemannian product ${S}^{n-1}\left(r\right)\times {S}^{1}\left(\sqrt{1-{r}^{2}}\right)$ in ${S}^{n+1}\left(1\right)$, or the image of the conformal diffeomorphism $\sigma $ of the standard cylinder ${S}^{n-1}\left(1\right)\times \mathbf{R}$ in ${\mathbf{R}}^{n+1}$, or the image of the conformal diffeomorphism $\tau $ of the Riemannian product ${S}^{n-1}\left(r\right)\times {\mathbf{H}}^{1}\left(\sqrt{1+{r}^{2}}\right)$in ${\mathbf{H}}^{n+1}$, or $\mathbf{x}$ is locally Möbius equivalent to the Veronese surface in ${S}^{4}\left(1\right)$. When $p=1$, our pinching condition is the same as in Main Theorem of Hu and Li [6], in which they assumed that $M$ is compact and the Möbius scalar curvature $n(n-1)R$ is constant. Secondly, we consider the Möbius sectional curvature of the immersion $\mathbf{x}$. We obtained that, for an $n$-dimensional compact submanifold $\mathbf{x}:M\mapsto {S}^{n+p}\left(1\right)$ without umbilical points and with vanishing form $\mathrm{\Phi}$, if the Möbius scalar curvature $n(n-1)R$ of the immersion $\mathbf{x}$ is constant and the Möbius sectional curvature $K$ of the immersion $\mathbf{x}$ satisfies $K\ge 0$ when $p=1$ and $K>0$ when $p>1$. Then, $\mathbf{x}$ is Möbius equivalent to either the Riemannian product ${S}^{k}\left(r\right)\times {S}^{n-k}\left(\sqrt{1-{r}^{2}}\right)$, for $k=1,2,\cdots ,n-1$, in ${S}^{n+1}\left(1\right)$; or $\mathbf{x}$ is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in ${S}^{n+p}\left(1\right)$.

## Citation

Qing-Ming CHENG. Shichang SHU. "A Möbius characterization of submanifolds." J. Math. Soc. Japan 58 (3) 903 - 925, July, 2006. https://doi.org/10.2969/jmsj/1156342043

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