[Paper Review] About the Schouten-Weyl tensor on 3-dimensional Lorenzian Lie groups

Олеся Павловна Хромова, P. N. Klepikov+2 more|arXiv (Cornell University)|Aug 22, 2017

Advanced Differential Geometry Research2 references3 citations

TL;DR

This paper investigates the Schouten–Weyl tensor on 3-dimensional Lie groups equipped with left-invariant Lorentzian metrics, focusing on cases where the tensor has zero squared length (isotropic) or is almost harmonic (zero curl and divergence). Using explicit computations in pseudo-orthonormal bases, the authors classify Lie algebras and vector fields for which the Schouten–Weyl tensor and its contractions exhibit harmonicity or isotropy, yielding specific algebraic conditions on structure constants and vector components, with key results in Lie algebras $\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{A}_3$, and $\mathcal{A}_4$. The work extends prior studies on harmonic Weyl tensors to the Lorentzian setting in dimension three.

ABSTRACT

The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric Lie groups with almost harmonic (i.e. with zero curl and divergence) Schouten-Weyl tensor are investigated. In addition, the question about the harmonicity of contraction of the Schouten-Weyl tensor is considered.

Motivation & Objective

• To classify 3-dimensional metric Lie groups with left-invariant Lorentzian metrics for which the Schouten–Weyl tensor is isotropic (zero squared length).
• To identify Lie groups where the Schouten–Weyl tensor is almost harmonic, i.e., has zero curl and divergence.
• To investigate the harmonicity of the contraction of the Schouten–Weyl tensor with left-invariant vector fields.
• To determine conditions under which the resulting tensor from contraction is harmonic, using curl and divergence operators.

Proposed method

• The study employs a pseudo-orthonormal frame field on unimodular 3D Lie groups to express the Schouten–Weyl tensor in terms of structure constants and metric components.
• The squared length of the Schouten–Weyl tensor is computed via $\|SW\|^2 = SW_{ijk}SW^{ijk}$, and isotropy is defined by $\|SW\|^2 = 0$ with $SW \neq 0$.
• Harmonicity of the Schouten–Weyl tensor and its contractions is analyzed using curl and divergence operators defined via covariant derivatives of the tensor components.
• For contraction with a left-invariant vector field $V^k$, the tensor $w_{ij} = V^k SW_{kij}$ is formed, and its curl and divergence are computed to test harmonicity.
• The authors solve systems of polynomial equations derived from $\mathrm{curl}(w) = 0$ and $\mathrm{div}(w) = 0$ to find solutions in terms of structure constants and vector field components.
• Harmonic vector fields are identified by solving $V^i_{,j} - V^j_{,i} = 0$ and $V^i_{,i} = 0$, which reduce to algebraic conditions on the components and structure constants.

Experimental results

Research questions

• RQ1 For which 3-dimensional unimodular Lie groups with left-invariant Lorentzian metrics is the Schouten–Weyl tensor isotropic?
• RQ2 Which 3D Lie groups admit an almost harmonic Schouten–Weyl tensor (zero curl and divergence)?
• RQ3 Under what conditions is the contraction of the Schouten–Weyl tensor with a left-invariant vector field harmonic?
• RQ4 Which specific Lie algebras ($\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{A}_3$, $\mathcal{A}_4$) support harmonic or isotropic Schouten–Weyl tensors?
• RQ5 What are the explicit algebraic relations between structure constants and vector field components that yield harmonic contractions?

Key findings

• For Lie algebra $\mathcal{A}_1$, solutions with $V = (V^1, V^2, V^3)$, $\lambda_1 = -2$, $\lambda_2 = 0$ yield a harmonic contraction of the Schouten–Weyl tensor.
• In $\mathcal{A}_1$, a second solution exists with $V = (0, V^2, V^3)$, $\lambda_2 = \frac{(V^3)^2 - (V^2)^2}{2V^2V^3}$, and $\lambda_1$ expressed via complex radicals involving $F$, $H$, and $Q$ functions of $V^2$ and $V^3$, excluding a specific ratio of $V^2/V^3$.
• For $\mathcal{A}_1$, the solution $V = (0, V^2, V^2)$ with $\lambda_1 \in \mathbb{R} \setminus \{0\}$, $\lambda_2 = 0$ yields a harmonic contraction.
• The solution $V = (V^1, 0, 0)$ with $\lambda_1 = 0$, $\lambda_2 \neq 0$ does not yield a harmonic contraction, as it fails the harmonic vector field condition.
• For $\mathcal{A}_3$, the solution $V = (0, -V^3, V^3)$ with $\lambda = 0$ produces a harmonic contraction of the Schouten–Weyl tensor.
• For $\mathcal{A}_3$, a second solution exists with $V^1$, $V^2$ expressed as rational functions of $V^3$ and $\lambda$, where $\lambda \approx \pm 89.072$, and the contraction is not harmonic due to violation of the harmonic vector field condition.