Journal of the
Korean Mathematical Society JKMS

The Iwasawa decomposition $NAK$ of the Lie group $G=\text{SO}_0(2,1)$ with a left invariant metric produces Riemannian submersions $ G\ra N\bs G$,\ $G\ra A\bs G$,\ $G\ra K\bs G$,\ and $G\ra \NA\bs G$. 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

Supported by: The author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2018R1D1A1A02047995).