[Paper Review] On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups

Akira Kubo, Kensuke Onda+2 more|arXiv (Cornell University)|Sep 28, 2015

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper develops a generalized Milnor frame procedure for left-invariant pseudo-Riemannian metrics on Lie groups using the moduli space of such metrics, enabling classification of curvature properties. It proves that all left-invariant pseudo-Riemannian metrics of arbitrary signature on real hyperbolic space Lie groups have constant sectional curvature, generalizing known results for Lorentzian and Riemannian cases.

ABSTRACT

In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. As one of applications, we show that any left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.

Motivation & Objective

• To formulate a general procedure for constructing Milnor-type frames for left-invariant pseudo-Riemannian metrics on Lie groups.
• To extend the moduli space approach—previously used for Riemannian metrics—to the pseudo-Riemannian setting.
• To analyze curvature properties of left-invariant pseudo-Riemannian metrics on specific Lie groups, particularly real hyperbolic space Lie groups.
• To demonstrate that all such metrics on real hyperbolic space Lie groups have constant sectional curvature, regardless of signature.
• To clarify differences between Riemannian and pseudo-Riemannian cases through structural and curvature analysis.

Proposed method

• The method is based on the moduli space of left-invariant pseudo-Riemannian metrics, defined as the orbit space of the action of ℝ×Aut(𝔤) on the space of metrics of signature (p,q).
• A generalized Milnor frame is constructed by finding representatives in the moduli space that simplify bracket relations of the Lie algebra.
• The procedure uses a set of canonical representatives, such as 𝔘* = {I₃ − λE₃,₁ | λ = 0,1,2}, to classify possible bracket structures under pseudo-orthonormal bases.
• The method applies to the Lie algebra 𝔤_ℝ𝐻ⁿ of the real hyperbolic space Lie group G_ℝ𝐻ⁿ, with bracket relations [e₁,eⱼ] = eⱼ for j ≥ 2.
• It leverages the fact that 𝔐_(p,q)(G) ≅ GLₙ(ℝ)/O(p,q) is a pseudo-Riemannian symmetric space when p,q ≥ 1, enabling classification via group actions.
• The construction is validated by showing that the resulting bracket relations depend only on a discrete parameter λ ∈ {0,1,2}, simplifying curvature computation.

Experimental results

Research questions

• RQ1 Can a generalized Milnor frame procedure be developed for left-invariant pseudo-Riemannian metrics, analogous to the known Riemannian case?
• RQ2 Do left-invariant pseudo-Riemannian metrics on real hyperbolic space Lie groups have constant sectional curvature for arbitrary signatures?
• RQ3 How do the curvature properties of left-invariant pseudo-Riemannian metrics differ from their Riemannian counterparts on the same Lie groups?
• RQ4 What role does the moduli space structure ℝ×Aut(𝔤) \\ 𝔐_(p,q)(G) play in classifying such metrics?
• RQ5 Can the discrete parameter λ ∈ {0,1,2} fully classify the bracket relations of pseudo-orthonormal bases under arbitrary signatures?

Key findings

• All left-invariant pseudo-Riemannian metrics of signature (p,q) on the real hyperbolic space Lie group G_ℝ𝐻ⁿ have constant sectional curvature.
• For any real number, there exists a left-invariant pseudo-Riemannian metric of signature (p,q) on G_ℝ𝐻ⁿ realizing that number as its constant sectional curvature.
• The bracket relations of the generalized Milnor frame depend only on a discrete parameter λ ∈ {0,1,2}, simplifying curvature analysis.
• The Lorentzian case (q=1) is recovered as a special case, providing an alternative proof of Nomizu’s result on constant curvature.
• The metric corresponding to λ=1 is flat, while λ=0 and λ=2 yield non-Einstein but algebraic Ricci soliton metrics.
• The classification via λ ∈ {0,1,2} establishes a complete and finite parameterization of possible pseudo-orthonormal frame structures for the Lie algebra of G_ℝ𝐻ⁿ.