In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an $ n $-dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an $ (n-1) $-dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.

中文翻译：

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一类多重扭曲乘积半黎曼爱因斯坦度量

在本文中，我们描述了当基数与n维维伪欧氏空间共形时，多重扭曲积半黎曼流形的特征。我们证明了翘曲积半黎曼流形为爱因斯坦流形的一些条件，该条件在$（n-1）$维翻译组的作用下是不变的。此后，当纤维为Ricci-flat时，我们将此结果应用于Ricci-flat多重扭曲产品空间的情况。我们还讨论了在具有null like向量的相同作用下，无限多个Ricci-flat乘法翘曲积空间的存在。