  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 $f\colon M^n\to \R^m$ 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.
