## Journal of the Mathematical Society of Japan

### A Möbius characterization of submanifolds

#### Abstract

In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere $S^{n+p}(1)$. First of all, we proved that, for an $n$-dimensional $(n\geq 2)$ submanifold $\mathbf x:M\mapsto S^{n+p}(1)$ without umbilical points and with vanishing Möbius form $\Phi$, if $(n-2)||\tilde{\mathbf{A}}|| \leq\sqrt{\frac{n-1}n} \left\{ nR-\frac{1}{n}[(n-1)\left( 2-\frac{1}{p} \right)-1] \right\}$ is satisfied, then, $\mathbf x$ is Möbius equivalent to an open part of either the Riemannian product $S^{n-1}(r)\times S^{1}\left(\sqrt{1-r^2}\right)$ in $S^{n+1}(1)$, or the image of the conformal diffeomorphism $\sigma$ of the standard cylinder $S^{n-1}(1)\times \mathbf{R}$ in $\mathbf{R}^{n+1}$, or the image of the conformal diffeomorphism $\tau$ of the Riemannian product $S^{n-1}(r)\times \mathbf{H}^{1}\left(\sqrt{1+r^2}\right)$in $\mathbf{H}^{n+1}$, or $\mathbf x$ is locally Möbius equivalent to the Veronese surface in $S^4(1)$. 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 $\mathbf x:M\mapsto S^{n+p}(1)$ without umbilical points and with vanishing form $\Phi$, 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 $K\geq 0$ when $p=1$ and $K>0$ when $p>1$. Then, $\mathbf x$ is Möbius equivalent to either the Riemannian product $S^k(r)\times S^{n-k}\left(\sqrt{1-r^2}\right)$, for $k=1, 2, \cdots, n-1$, in $S^{n+1}(1)$; or $\mathbf x$ is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in $S^{n+p}(1)$.

#### Article information

Source
J. Math. Soc. Japan, Volume 58, Number 3 (2006), 903-925.

Dates
First available in Project Euclid: 23 August 2006

https://projecteuclid.org/euclid.jmsj/1156342043

Digital Object Identifier
doi:10.2969/jmsj/1156342043

Mathematical Reviews number (MathSciNet)
MR2254416

Zentralblatt MATH identifier
1102.53009

#### Citation

CHENG, Qing-Ming; SHU, Shichang. A Möbius characterization of submanifolds. J. Math. Soc. Japan 58 (2006), no. 3, 903--925. doi:10.2969/jmsj/1156342043. https://projecteuclid.org/euclid.jmsj/1156342043

#### References

• M. A. Akivis and V. V. Goldberg, Conformal differential geometry and its generalizations, Wiley, New York, 1996.
• M. A. Akivis and V. V. Goldberg, A conformal differential invariant and the conformal rigidity of hypersurfaces, Proc. Amer. Math. Soc., 125 (1997), 2415–2424.
• R. L. Bryant, Minimal surfaces of constant curvature in $S^n$, Trans. Amer. Math. Soc., 290 (1985), 259–271.
• Q.-M. Cheng, Submanifolds with constant scalar curvature, Proc. Royal Soc. Edinburgh, 132A (2002), 1163–1183.
• S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifols of a sphere with second fundamental form of constant length, Berlin, New York: Shing-Shen Chern Selected Papers, 1978, 393–409.
• Z. J. Hu and H. Li, Submanifolds with constant Möbius scalar curvature in $S^n$, Manuscripta Math., 111 (2003), 287–302.
• H. Li, H. L. Liu, C. P. Wang and G. S. Zhao, Möbius isoparametric hypersurface in $S^{n+1}$ with two distinct principal curvatures, Acta Math. Sinica, English Series, 18 (2002), 437–446.
• H. Li, C. P. Wang and F. Wu, Möbius characterization of Veronese surfaces in $S^{n}$, Math. Ann., 319 (2001), 707–714.
• H. L. Liu, C. P. Wang and G. S. Zhao, Möbius isotropic submanifolds in $S^n$, Tôhoku Math. J., 53 (2001), 553–569.
• W. Santos, Submanifolds with parallel mean curvature vector in spheres, Tôhoku Math. J., 46 (1994), 403–415.
• C. P. Wang, Möbius geometry of submanifolds in $S^n$, Manuscripta Math., 96 (1998), 517–534.