We prove that conformal immersion of a Riemannian product $$M_0^{n_0}\times M_1^{n_1}$$ M 0 n 0 × M 1 n 1 as a hypersurface in a Euclidean space must be an extrinsic product of immersions, under the assumption that $$n_0, n_1 \ge 2$$ n 0 , n 1 ≥ 2 and that $$M^{n_0}_0\times M^{n_1}_1$$ M 0 n 0 × M 1 n 1 is not conformally flat. We also state a similar theorem for an arbitrary number of factors, more precisely, a conformal immersion $$f:M^{n_0}_0 \times \cdots \times M^{n_k}_k \rightarrow {{\mathbb {R}}}^{n+k}$$ f : M 0 n 0 × ⋯ × M k n k → R n + k must be an extrinsic product of immersions if one of the factors admits a plane with vanishing curvature and the remaining factors are not flat.

中文翻译：

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黎曼乘积的保形浸入低维数

我们证明了作为欧几里得空间中的超曲面的黎曼积 $$M_0^{n_0}\times M_1^{n_1}$$ M 0 n 0 × M 1 n 1 的保形浸入必须是浸入的外积，在假设 $$n_0, n_1 \ge 2$$ n 0 , n 1 ≥ 2 并且 $$M^{n_0}_0\times M^{n_1}_1$$ M 0 n 0 × M 1 n 1 是不保形平坦。我们还为任意数量的因子陈述了一个类似的定理，更准确地说，保形浸入 $$f:M^{n_0}_0 \times \cdots \times M^{n_k}_k \rightarrow {{\mathbb {R} }}^{n+k}$$ f : M 0 n 0 × ⋯ × M knk → R n + k 必须是浸入的外积，如果其中一个因子允许一个具有消失曲率的平面而其余因子不存在平坦的。