The aim of this paper is to classify the cohomogeneity one conformal actions on the 3-dimensional Einstein universe $$\mathbb {E}{\mathrm {in}}^{1,2}$$ E in 1 , 2 , up to orbit equivalence. In a recent paper (Hassani in C R Acad Sci Paris Ser I 355:1133–1137, 2017. https://doi.org/10.1016/j.crma.2017.10.003 ), we studied the unique (up to conjugacy) irreducible action of $${\mathrm {PSL}}(2,\mathbb {R})$$ PSL ( 2 , R ) on $$\mathbb {E}{\mathrm {in}}^{1,2}$$ E in 1 , 2 and we showed that the action is of cohomogeneity one. In the present paper, we determine all the subgroups of $${\mathrm {Conf}}(\mathbb {E}{\mathrm {in}}^{1,2})$$ Conf ( E in 1 , 2 ) , up to conjugacy, acting reducibly and with cohomogeneity one on $$\mathbb {E}{\mathrm {in}}^{1,2}$$ E in 1 , 2 . We show that any cohomogeneity one reducible action on $$\mathbb {E}{\mathrm {in}}^{1,2}$$ E in 1 , 2 induces a fixed point in the 4-dimensional projective space $$\mathbb {R}\mathbb {P}^4$$ R P 4 . Also, we describe all the codimension one induced orbits by these actions.

中文翻译：

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三维爱因斯坦宇宙中的同质性一作用

本文的目的是对 3 维爱因斯坦宇宙 $$\mathbb {E}{\mathrm {in}}^{1,2}$$ E in 1 , 2 , up to轨道等效。在最近的一篇论文中（Hassani in CR Acad Sci Paris Ser I 355:1133–1137, 2017. https://doi.org/10.1016/j.crma.2017.10.003），我们研究了独特的（直到共轭）不可约$${\mathrm {PSL}}(2,\mathbb {R})$$ PSL ( 2 , R ) 对 $$\mathbb {E}{\mathrm {in}}^{1,2}$ 的作用$ E 在 1 , 2 中，我们表明该动作是同质性的。在本文中，我们确定 $${\mathrm {Conf}}(\mathbb {E}{\mathrm {in}}^{1,2})$$ Conf ( E in 1 , 2 ) 的所有子群，直到共轭，在 $$\mathbb {E}{\mathrm {in}}^{1,2}$$ E in 1 , 2 上可还原且具有同质性。我们证明了任何同质性对 $$\mathbb {E}{\mathrm {in}}^{1, 2}$$ E in 1 , 2 在 4 维射影空间 $$\mathbb {R}\mathbb {P}^4$$ RP 4 中导出不动点。此外，我们描述了通过这些动作引起的所有辅维一轨道。