[Paper Review] Left invariant semi Riemannian metrics on quadratic Lie groups

Shirley Bromberg, Alberto Abad Medina|arXiv (Cornell University)|Mar 7, 2011

Geometric Analysis and Curvature Flows16 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for the flatness of left-invariant semi-Riemannian metrics on quadratic Lie groups, proving that such metrics are always complete. It shows non-Abelian quadratic Lie groups cannot admit flat Riemannian or Lorentzian metrics, but every 3-step nilpotent quadratic Lie group admits a flat left-invariant semi-Riemannian metric, with infinitely many non-isometric such metrics existing when the dimension exceeds 8.

ABSTRACT

To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannian metric. We give a useful necessary and sufficient condition that guaranties the flatness of a left invariant semi Riemannian metric defined on a quadratic Lie group. All these semi Riemannian metrics are complete. We show that there are no Riemannian or Lorentzian flat left invariant metrics on non Abelian quadratic Lie groups, and that every quadratic 3 step nilpotent Lie group admits a flat left invariant semi Riemannian metric. The case of quadratic 2 step nilpotent Lie groups is also addressed.

Motivation & Objective

• To determine the conditions under which a left-invariant semi-Riemannian metric on a quadratic Lie group is flat.
• To resolve the open problem of classifying Lie groups admitting flat (and complete) left-invariant semi-Riemannian metrics.
• To analyze the existence and classification of flat left-invariant semi-Riemannian metrics on nilpotent quadratic Lie groups, especially 2-step and 3-step nilpotent cases.
• To establish a connection between solutions of the classical Yang-Baxter equation and flat left-invariant semi-Riemannian metrics on dual Lie groups.

Proposed method

• Utilizes the Levi-Civita product and curvature tensor analysis to derive conditions for flatness of left-invariant semi-Riemannian metrics on unimodular Lie groups.
• Applies the theory of bi-invariant semi-Riemannian metrics on quadratic (orthogonal) Lie groups, where the metric is preserved under both left and right translations.
• Analyzes Jacobi fields and conjugate points on quadratic Lie groups using the reflection property of the metric on the Lie algebra.
• Constructs flat left-invariant semi-Riemannian metrics via symmetric linear isomorphisms $ u $ on the Lie algebra that preserve the descending central series.
• Employs the Vinberg-Elashvili classification of 3-forms to show the existence of infinitely many non-isometric flat metrics when $ \dim V \geq 9 $.
• Uses the duality between solutions of the classical Yang-Baxter equation and flat metrics on dual Lie groups to generate new examples.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a left-invariant semi-Riemannian metric on a quadratic Lie group to be flat?
• RQ2 Can non-Abelian quadratic Lie groups admit flat Riemannian or Lorentzian left-invariant metrics?
• RQ3 Which nilpotent quadratic Lie groups admit flat left-invariant semi-Riemannian metrics, and how many non-isometric such metrics exist?
• RQ4 What is the relationship between solutions of the classical Yang-Baxter equation and the existence of flat left-invariant semi-Riemannian metrics on dual Lie groups?

Key findings

• A left-invariant semi-Riemannian metric on a quadratic Lie group is flat if and only if the associated symmetric linear isomorphism $ u $ on the Lie algebra satisfies a specific algebraic condition involving the Levi-Civita product.
• All flat left-invariant semi-Riemannian metrics on quadratic Lie groups are geodesically complete.
• No non-Abelian quadratic Lie group admits a flat left-invariant Riemannian or Lorentzian metric.
• Every 3-step nilpotent quadratic Lie group admits at least one flat left-invariant semi-Riemannian metric.
• For 2-step nilpotent quadratic Lie groups of 0 corank, there exist infinitely many non-isometric flat left-invariant semi-Riemannian metrics when $ \dim G > 8 $.
• When $ \dim V \geq 9 $, the Vinberg-Elashvili classification ensures the existence of infinitely many non-conjugate 3-forms, leading to infinitely many non-isometric flat metrics on the associated Lie group.