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<p>We describe all left-invariant Bach-flat Lorentz metrics on non-solvable four-dimensional Lie groups, showing that they are conformally Einstein if and only if they are locally conformally flat. As a consequence we show the existence of strictly Bach-flat left-invariant metrics on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10183_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(SU(2)\times \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> whose restriction to <i>SU</i>(2) may be positive definite, Lorentzian or degenerate.</p>

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Conformally Einstein Lorentzian Lie Groups: The Non-solvable Case

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月4日

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  • E. Calviño-Louzao,
  • E. García-Río,
  • I. Gutiérrez-Rodríguez,
  • R. Vázquez-Lorenzo

摘要

We describe all left-invariant Bach-flat Lorentz metrics on non-solvable four-dimensional Lie groups, showing that they are conformally Einstein if and only if they are locally conformally flat. As a consequence we show the existence of strictly Bach-flat left-invariant metrics on \(SU(2)\times \mathbb {R}\) S U ( 2 ) × R whose restriction to SU(2) may be positive definite, Lorentzian or degenerate.