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<p>Let <i>G</i> be a Lie group equipped with a left-invariant Riemannian metric. Let <i>K</i> be a semisimple and normal subgroup of <i>G</i> generating a left-invariant conformal foliation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10015_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> on <i>G</i>. We then show that the foliation <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10015_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> is Riemannian and minimal. This means that locally the leaves of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10015_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> are fibres of a harmonic morphism. We also prove that if the metric restricted to <i>K</i> is biinvariant then <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10015_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> is totally geodesic.</p>

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Harmonic Morphisms and Minimal Conformal Foliations on Lie Groups

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月30日

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  • Sigmundur Gudmundsson,
  • Thomas Jack Munn

摘要

Let G be a Lie group equipped with a left-invariant Riemannian metric. Let K be a semisimple and normal subgroup of G generating a left-invariant conformal foliation \(\mathcal {F}\) F on G. We then show that the foliation \(\mathcal {F}\) F is Riemannian and minimal. This means that locally the leaves of \(\mathcal {F}\) F are fibres of a harmonic morphism. We also prove that if the metric restricted to K is biinvariant then \(\mathcal {F}\) F is totally geodesic.