On Complete Conformally Flat Submanifolds with Nullity in Euclidean Space

Abstract

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f:M^n\rightarrow \mathord {\mathbb {R}}^m$ be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then $M^n$ is flat and f is a cylinder over a flat submanifold. (2) If the scalar curvature of $M^n$ is non-negative and the index of relative nullity is positive, then f is a cylinder over a submanifold with constant non-negative sectional curvature. (3) If the scalar curvature of $M^n$ is non-zero and the index of relative nullity is constant and equal to one, then f is a cylinder over a $(n-1)$-dimensional submanifold with non-zero constant sectional curvature.