Naturally reductive pseudo-Riemannian Lie groups in low dimensions

Abstract

This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on $\Heis_3(\RR)$ by isometries and we finally study some geometrical features on these spaces.

Keywords: Pseudo-Riemannian spaces, naturally reductive, Lie groups, Heisenberg group

References in corpus (4)

• Supersymmetry and homogeneity of M-theory backgrounds
• Left-invariant Lorentzian flat metrics on Lie groups
• Naturally reductive pseudo-Riemannian spaces
• Ricci flat left invariant Lorentzian metrics on 2-step nilpotent Lie groups

Cited by in corpus (4)

• Witt pseudo-Riemannian connection
• Lorentzian compact manifolds: isometries and geodesics
• Four-dimensional naturally reductive pseudo-Riemannian spaces
• Isometric actions on pseudo-Riemannian nilmanifolds