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 $⟨\cdot ,\cdot {⟩}^{\ast }$ and the Moebius second fundamental form $\beta$. In [11] was introduced the class of umbilic-free Moebius semi-parallel submanifolds of the unit sphere, which means that $\bar{R}\cdot \beta =0$, where $\bar{R}$ is the van der Waerden-Bortolotti curvature operator associated to $⟨\cdot ,\cdot {⟩}^{\ast }$. 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.

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