In this article, we obtain that compact simple Lie groups <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1614_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="172" /> </InlineMediaObject> <EquationSource Format="TEX">$Sp(n)(n=4k+6\ge 26)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>6</mn> <mo>≥</mo> <mn>26</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> admit at least two new non-naturally reductive <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1614_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="348" /> </InlineMediaObject> <EquationSource Format="TEX">$Ad(Sp(k+2)\times Sp(k+2)\times Sp(k+2)\times Sp(k))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi>S</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi>S</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi>S</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-invariant Einstein metrics, and we prove that these Einstein metrics are not geodesic orbit metrics. Furthermore, we construct two left invariant non-geodesic orbit Einstein–Randers metrics on Lie group Sp(n).

New non-geodesic orbit Einstein metrics on Sp(n)

期刊论文
发表日期：

2025年2月5日

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Wenyan Luo,
Ju Tan,
Na Xu

摘要

In this article, we obtain that compact simple Lie groups $Sp(n)(n=4k+6\ge 26)$ $S p (n) (n = 4 k + 6 \geq 26)$ admit at least two new non-naturally reductive $Ad(Sp(k+2)\times Sp(k+2)\times Sp(k+2)\times Sp(k))$ $A d (S p (k + 2) \times S p (k + 2) \times S p (k + 2) \times S p (k))$ -invariant Einstein metrics, and we prove that these Einstein metrics are not geodesic orbit metrics. Furthermore, we construct two left invariant non-geodesic orbit Einstein–Randers metrics on Lie group Sp(n).