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

中文翻译：

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欧几里得空间中带零的完全共形平坦子流形

在本笔记中，我们研究了欧几里得空间的共形平坦子流形，其相对无效指数为正。令 $$M^n$$ M n 是一个完整的共形平面流形，并令 $$f:M^n\rightarrow \mathord {\mathbb {R}}^m$$ f : M n → R m 是一个等距浸没。我们证明以下结果： (1) 如果相对无效指数至少为 2，则 $$M^n$$ M n 是平坦的，f 是平坦子流形上的圆柱体。(2) 如果$$M^n$$M n 的标量曲率非负且相对无效指数为正，则f 是具有恒定非负截面曲率的子流形上的圆柱。(3) 若$$M^n$$ M n 的标量曲率非零且相对无效指数为常数且等于1，则f 是$$(n-1)$$ 上的圆柱( n - 1 ) 维子流形，具有非零恒定截面曲率。