In this study, we consider the generalized <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Ricci soliton (RS, in short) within the framework of a contact metric manifold <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M^{2n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\varphi , \xi , \eta , g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as a contact metric structure satisfying certain conditions on the potential vector field. First, we establish that a compact contact metric admitting a generalized <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-RS with potential vector V as an infinitesimal harmonic transformation (IHT, in short) is <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Einstein provide <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\pounds _V(div V)\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">£</mi> <mi>V</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mi>i</mi> <mi>v</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Next, we prove that a generalized <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-RS whose potential vector field V is parallel with the Reeb vector field and satisfies <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$Q\xi = (Trl)\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mi>ξ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mi>r</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mi>ξ</mi> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Einstein. Finally, we have studied the contact metric manifold with <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$Q\varphi = \varphi Q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mi>φ</mi> <mo>=</mo> <mi>φ</mi> <mi>Q</mi> </mrow> </math></EquationSource> </InlineEquation> and admitting a generalized <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-RS when its non-zero potential vector field V is a contact vector field, or a projective vector field.

SOME RESULTS ON GENERALIZED $\eta $-RICCI SOLITONS

期刊论文
发表日期：

2025年5月7日

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Amalendu Ghosh

摘要

In this study, we consider the generalized $\eta $ $η$ -Ricci soliton (RS, in short) within the framework of a contact metric manifold $M^{2n+1}$ $M^{2 n + 1}$ with $(\varphi , \xi , \eta , g)$ $(φ, ξ, η, g)$ as a contact metric structure satisfying certain conditions on the potential vector field. First, we establish that a compact contact metric admitting a generalized $\eta $ $η$ -RS with potential vector V as an infinitesimal harmonic transformation (IHT, in short) is $\eta $ $η$ -Einstein provide $\pounds _V(div V)\ge 0$ $£_{V} (d i v V) \geq 0$ . Next, we prove that a generalized $\eta $ $η$ -RS whose potential vector field V is parallel with the Reeb vector field and satisfies $Q\xi = (Trl)\xi $ $Q ξ = (T r l) ξ$ is $\eta $ $η$ -Einstein. Finally, we have studied the contact metric manifold with $Q\varphi = \varphi Q$ $Q φ = φ Q$ and admitting a generalized $\eta $ $η$ -RS when its non-zero potential vector field V is a contact vector field, or a projective vector field.

SOME RESULTS ON GENERALIZED \(\eta \)-RICCI SOLITONS

摘要