<p>In this paper, we consider <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\eta\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Ricci soliton on <i>K</i>-contact manifold. First, we prove that a <i>K</i>-contact metric representing an <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\eta\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Ricci soliton is either Einstein, or <i>D</i>-homothetically fixed <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\eta\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Einstein provided the Ricci operator <i>Q</i> commutes with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>. Next, we prove that a compact <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\eta\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Ricci soliton (<InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">\(g,V,\lambda , \mu\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>) on a <i>K</i>-contact manifold <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(M^{2n+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\eta\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Einstein (trivial) with Killing potential vector field. Finally, a couple of results are proved on contact metric manifold admitting <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_932_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\eta\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Ricci soliton when its non-zero potential vector field is parallel with the Reeb vector field.</p>

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Certain contact metric as \(\eta\)-Ricci soliton

  • Amalendu Ghosh

摘要

In this paper, we consider \(\eta\) η -Ricci soliton on K-contact manifold. First, we prove that a K-contact metric representing an \(\eta\) η -Ricci soliton is either Einstein, or D-homothetically fixed \(\eta\) η -Einstein provided the Ricci operator Q commutes with \(\varphi\) φ . Next, we prove that a compact \(\eta\) η -Ricci soliton ( \(g,V,\lambda , \mu\) g , V , λ , μ ) on a K-contact manifold \(M^{2n+1}\) M 2 n + 1 is \(\eta\) η -Einstein (trivial) with Killing potential vector field. Finally, a couple of results are proved on contact metric manifold admitting \(\eta\) η -Ricci soliton when its non-zero potential vector field is parallel with the Reeb vector field.