We classify the effective and transitive actions of a Lie group G on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that G is a closed, connected Lie subgroup of SO₀(n–r; r+1), the connected component of the indefinite special orthogonal group. Assuming additionally that G acts completely reducibly on ℝ ^{n +1} , we also obtain that any G-homogeneous Einstein pseudo-Riemannian metric on a real, complex or quaternionic pseudo-hyperbolic space, or on a para-complex or para-quaternionic projective space is homothetic to either the canonical metric or the Einstein metric of the canonical variation of a Hopf pseudo-Riemannian submersion.

中文翻译：

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伪双曲空间上的同质爱因斯坦矩阵的分类

在G是SO ₀（n – r的一个封闭的，连通的李子群）的假设下，我们将李群G对n维非简并双曲面（也称为实伪双曲空间）的有效和传递作用进行分类。; r +1），不定特殊正交群的连接分量。假设另外的是G ^作用完全reducibly上ℝ ^{Ñ 1}，我们也获得任何ģ实，复或四元拟双曲空间上，或在准复数或准四元射影射影空间上的齐次爱因斯坦拟黎曼度量与Hopf拟正则变体的规范度量或爱因斯坦度量是同质的-黎曼浸没。