Tohoku Mathematical Journal
2001
December
SECOND SERIES VOL. 53, NO. 4

Tohoku Math. J.
53 (2001), 553-569

Title MÖBIUS ISOTROPIC SUBMANIFOLDS IN $\boldsymbol{S}^n$

Author Huili Liu, Changping Wang and Guosong Zhao

(Received January 5, 2000, revised February 5, 2001)
Abstract. Let $x:\boldsymbol{M}^m \to \boldsymbol{S}^n$ be a submanifold in the $n$-dimensional sphere $\boldsymbol{S}^n$ without umbilics. Two basic invariants of $x$ under the Möbius transformation group in $\boldsymbol{S}^n$ are a 1-form ${\phi}$ called the Möbius form and a symmetric $(0,2)$ tensor $\bf A$ called the Blaschke tensor. $x$ is said to be Möbius isotropic in $S^n$ if ${\phi}\equiv 0$ and ${\bf A}={\lambda} dx\cdot dx$ for some smooth function ${\lambda}$. An interesting property for a Möbius isotropic submanifold is that its conformal Gauss map is harmonic. The main result in this paper is the classification of Möbius isotropic submanifolds in $\boldsymbol{S}^n$. We show that (i) if $\lambda >0$, then $x$ is Möbius equivalent to a minimal submanifold with constant scalar urvature in $\boldsymbol{S}^n$; (ii) if $\lambda=0$, then $x$ is Möbius equivalent to the pre-image of a stereographic projection of a minimal submanifold with constant scalar curvature in the $n$-dimensional Euclidean space $\boldsymbol{R}^n$; (iii) if $\lambda <0$, then $x$ is Möbius equivalent to the image of the standard conformal map $\tau: \boldsymbol{H}^n \to \boldsymbol{S}^n_+$ of a minimal submanifold with constant scalar curvature in the $n$-dimensional hyperbolic space $\boldsymbol{H}^n$. This result shows that one can use Möbius differential geometry to unify the three different classes of minimal submanifolds with constant scalar curvature in $\boldsymbol{S}^n$, $\boldsymbol{R}^n$ and $\boldsymbol{H}^n$.

2000 Mathematics Subject Classification. Primary 53A30; Secondary 53B25.
Key words and phrases. Möbius geometry, isotropic submanifold, minimal submanifold, scalar curvature.

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