Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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INFRA-SOLVMANIFOLDS OF Sol14

  • Received : 2014.12.27
  • Published : 2015.11.01
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Abstract

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$, and more generally, the crystallographic groups of $Sol_1^4$. The maximal compact subgroup of Isom($Sol_1^4$) is $D_4={\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$. We shall exhibit an infra-solvmanifold of $Sol_1^4$ whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; the trivial group, ${\mathbb{Z}}_2$ (5 families), ${\mathbb{Z}}_4$, ${\mathbb{Z}}_2{\times}{\mathbb{Z}}_2$ (5 families), and ${\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$ (2 families).

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References

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Cited by

  1. -MANIFOLDS pp.1446-8107, 2018, https://doi.org/10.1017/S1446788717000258
  2. Classification of closed manifolds with $${\mathrm{Sol}_1}^{4}$$Sol14-geometry pp.1572-9168, 2018, https://doi.org/10.1007/s10711-018-0354-1