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.