Riemannian submersions of $\text{SO}(2,1)$
J. Korean Math. Soc.
Published online April 13, 2021
Taechang Byun
Sejong University
Abstract : The Iwasawa decomposition $NAK$ of the Lie group $G=\text{SO}_0(2,1)$ with a left invariant metric produces Riemannian submersions $ G \rightarrow N \backslash G$, $G \rightarrow A \backslash G$, $G \rightarrow K \backslash G$, and $G \rightarrow NA \backslash G$.

For each of these, we calculate the curvature of the base space and the lifting of a simple closed curve to the total space $G$. Especially in the first case, the base space has a constant curvature 0; the holonomy displacement along a (null-homotopic) simple closed curve in the base space is determined only by the Euclidean area of the region surrounded by the curve.
Keywords : Holonomy displacement, Area form, Riemannian submersion, Principal bundle, Lie algebra, Nilpotent Lie group, Solvable Lie group
MSC numbers : 22E25, 53C30, 53C29, 52A38, 55M25
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