A Moebius characterization of Veronese surfaces in $S^n$

Abstract.

Let $M^m$ be an umbilic-free submanifold in $S^n$ with I and II as the first and second fundamental forms. An important Moebius invariant for $M^m$ in Moebius differential geometry is the so-called Moebius form $\Phi$, defined by $\Phi =-\rho^{-2}\sum_{i,\alpha}\{H^{\alpha}_{,i} +\sum_j(II^{\alpha}_{ij}-H^{\alpha}I_{ij})e_j(\log \rho)\{\omega_i\otimes e_{\alpha}$, where $\{e_i\}$ is a local basis of the tangent bundle with dual basis \{\omega_i\}\), $\{e_{\alpha}\}$ is a local basis of the normal bundle, $H=\sum_{\alpha}H^{\alpha}e_{\alpha}$ is the mean curvature vector and $\rho =\sqrt{m\over{m-1}}\|II-HI\|$. In this paper we prove that if $x: S^2\to S^n$ is an umbilics-free immersion of 2-sphere with vanishing Moebius form $\Phi$, then there exists a Moebius transformation $\tau: S^n\to S^n$ and a 2k-equator $S^{2k}\subset S^n$ with $2\le k\le [n/2]$ such that $\tau\circ x:S^2\to S^{2k}$ is the Veronese surface.