# On Complete Conformally flat submanifolds with nullity in Euclidean space

## TL;DR

This paper classifies complete conformally flat submanifolds in Euclidean space with positive nullity index, showing they are cylinders over flat or constant curvature submanifolds under various curvature conditions.

## Contribution

It provides a complete classification of such submanifolds based on nullity index and scalar curvature, extending previous results in conformal geometry.

## Key findings

- Nullity index at least two implies the submanifold is flat and cylindrical.
- Non-negative scalar curvature with positive nullity implies a cylindrical structure over a constant curvature submanifold.
- Non-zero scalar curvature with nullity one implies a cylindrical structure over a submanifold with non-zero constant sectional curvature.

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