Cyclic Riemannian Lie groups: description and curvatures
Fatima-Ezzahrae Abid, Said Benayadi, Mohamed Boucetta, Hamza El Ouali, Hicham Lebzioui
TL;DR
The paper advances the classification of cyclic Riemannian Lie groups by proving that any connected, simply-connected instance is isomorphic to $\mathbb{R}^r \times G(q,p,\Omega) \times (\widetilde{\mathrm{SL}(2,\mathbb{R})})^m$, with $G(q,p,\Omega)$ carrying a specific solvable group law and left-invariant metric $h_0$. It provides explicit structural details, including the rank condition on the $\Omega$ matrix and the form of the metric, and reduces the problem to solvable versus semisimple components involving $\widetilde{\mathrm{SL}(2,\mathbb{R})}$. The curvature analysis shows the scalar curvature is always negative for non-abelian cases, and it characterizes when cyclic groups have constant sectional curvature, negative Ricci curvature, parallel Ricci curvature, symmetry, or are Einstein, with a complete dimension-by-dimension listing up to dimension five. These results extend earlier work and yield practical models for constructing and studying cyclic Riemannian Lie groups and their geometric properties.
Abstract
A cyclic Riemannian Lie group is a Lie group $G$ equipped with a left-invariant Riemannian metric $h$ that satisfies $\oint_{X,Y,Z}h([X,Y],Z)=0$ for any left-invariant vector fields $X,Y,Z$. The initial concept and exploration of these Lie groups were presented in Monatsh. Math. \textbf{176} (2015), 219-239. This paper builds upon the results from the aforementioned study by providing a complete description of cyclic Riemannian Lie groups and an in-depth analysis of their various curvatures.