A Möbius characterization of submanifolds

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 $S^{n+p}\left(1\right)$.