Scholarlink

<p>In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics (<i>M</i>,&#xa0;<i>g</i>,&#xa0;<i>X</i>) with <i>X</i> a left-invariant vector field, which we call <i>totally left-invariant quasi-Einstein metrics</i>. We give a complete classification of nilpotent Lie groups admitting such metrics, proving that this occurs <i>if and only if the group is isomorphic to a Heisenberg Lie group</i>. For unimodular solvable Lie groups <i>S</i>, we show that the existence of a non-flat totally left-invariant quasi-Einstein metric forces the center of <i>S</i> to be one-dimensional. Furthermore, under the additional assumption that the adjoint action <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\operatorname {ad}_a\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mo>ad</mo> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> of <i>S</i> is a normal derivation, we obtain a full classification: these groups are standard and their nilradical must be a Heisenberg Lie algebra. As an application, we prove that the only near-horizon geometries on a compact nilmanifold are <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Gamma \backslash H_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="true">\</mo> </mrow> <msub> <mi>H</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( H_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is <i>n</i>-dimensional Heisenberg Lie group.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On Nilpotent and Solvable Quasi-Einstein Manifolds

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年7月6日

查看全文
  • Nazia Valiyakath

摘要

In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics (M, g, X) with X a left-invariant vector field, which we call totally left-invariant quasi-Einstein metrics. We give a complete classification of nilpotent Lie groups admitting such metrics, proving that this occurs if and only if the group is isomorphic to a Heisenberg Lie group. For unimodular solvable Lie groups S, we show that the existence of a non-flat totally left-invariant quasi-Einstein metric forces the center of S to be one-dimensional. Furthermore, under the additional assumption that the adjoint action \(\operatorname {ad}_a\) ad a of S is a normal derivation, we obtain a full classification: these groups are standard and their nilradical must be a Heisenberg Lie algebra. As an application, we prove that the only near-horizon geometries on a compact nilmanifold are \(\Gamma \backslash H_{n}\) Γ \ H n , where \( H_{n}\) H n is n-dimensional Heisenberg Lie group.