On Complete Conformally Flat Submanifolds with Nullity in Euclidean Space

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 -> 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.