Bulletin of the
Korean Mathematical Society BKMS

The M\"{o}bius homogeneous submanifold in $\mathbb{S}^{n+1}$ is an orbit of a subgroup of the M\"{o}bius transformation group of $\mathbb{S}^{n+1}$. In this note, We prove that a compact M\"{o}bius homogeneous submanifold in $\mathbb{S}^{n+1}$ is the image of a M\"{o}bius transformation of the isometric homogeneous submanifold in $\mathbb{S}^{n+1}$.

Keywords: M\"{o}bius transformation group, isometric transformation group, M\"{o}bius homogeneous hypersurfaces, homogeneous hypersurfaces

MSC numbers: 53A30, 53C30

Supported by: Authors are partially supported by the grant No.11471021 and No. 11571037 of NSFC