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 $\boldsymbol{S}^n$ 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 $\boldsymbol{H}^n$.
