Normally flat submanifolds with semi-parallel Moebius second fundamental form

TL;DR

This paper classifies certain flat submanifolds in spheres with semi-parallel Moebius second fundamental form, advancing understanding of their geometric structure in Moebius geometry.

Contribution

It provides a classification of umbilic-free isometric immersions with semi-parallel Moebius second fundamental form and flat normal bundle.

Findings

Classification of umbilic-free immersions with semi-parallel Moebius second fundamental form.

Identification of conditions for flat normal bundle in these submanifolds.

Extension of previous work on Moebius semi-parallel submanifolds.

Abstract

In Moebius geometry there are two important tensors associated to an umbilic-free immersion $f:M^{n}\to \mathbb{S}^{m}$, namely the Moebius metric $\langle \cdot, \cdot \rangle^{*}$ 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 $\langle \cdot, \cdot \rangle^{*}$. 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.