Normally flat submanifolds with semi-parallel Moebius second fundamental form

Abstract

In the Moebius geometry there are two important tensors associated to an umbilic-free immersion $f:M^{n}\to \mathbb{R}^{m}$ named the Moebius metric $\langle \cdot, \cdot \rangle^{*}$ and the Moebius second fundamental form $β$. In [11] was introduced the class of umbilic-free Moebius semi-parallel submanifolds of the unit sphere, which means that $\bar{R}\cdot β=0$, where $\bar{R}$ is the van der Waerden-Bortolotti curvature operator associated to $\langle \cdot, \cdot \rangle^{*}$. In this paper, we classify umbilic-free isometric immersions $f:M^{n}\to \mathbb{R}^{m}$ with semi-parallel Moebius second fundamental form and flat normal bundle.