<p>This study investigates the geometric properties of almost <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{\eta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">η</mi> </mrow> </math></EquationSource> </InlineEquation>-Ricci-Bourguignon solitons on para-Kenmotsu manifolds. First, it is shown that if the metric <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varvec{g}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">g</mi> </mrow> </math></EquationSource> </InlineEquation> on a para-Kenmotsu manifold admits an almost <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{\eta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">η</mi> </mrow> </math></EquationSource> </InlineEquation>-Ricci-Bourguignon soliton and the Reeb vector field <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varvec{\xi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">ξ</mi> </mrow> </math></EquationSource> </InlineEquation> preserves the function <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varvec{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> </math></EquationSource> </InlineEquation>, then the manifold <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varvec{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation> is Einstein. Next, we establish that if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varvec{g}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">g</mi> </mrow> </math></EquationSource> </InlineEquation> is an almost <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\varvec{\eta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">η</mi> </mrow> </math></EquationSource> </InlineEquation>-Ricci-Bourguignon soliton and the vector field <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\varvec{\xi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">ξ</mi> </mrow> </math></EquationSource> </InlineEquation> leaves both <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\varvec{\lambda + \rho r}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">λ</mi> <mo mathvariant="bold">+</mo> <mi mathvariant="bold-italic">ρ</mi> <mi mathvariant="bold-italic">r</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\varvec{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> </math></EquationSource> </InlineEquation> invariant, then the soliton reduces to an <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\varvec{\eta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">η</mi> </mrow> </math></EquationSource> </InlineEquation>-Ricci-Bourguignon soliton. Finally, we prove that if <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\varvec{g}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">g</mi> </mrow> </math></EquationSource> </InlineEquation> is a gradient almost <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\varvec{\eta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">η</mi> </mrow> </math></EquationSource> </InlineEquation>-Ricci-Bourguignon soliton, then <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\varvec{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation> is necessarily Einstein.</p>

GEOMETRIC CHARACTERIZATIONS OF ALMOST $\eta $-RICCI-BOURGUIGNON SOLITONS ON PARA-KENMOTSU MANIFOLDS

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发表日期：

2026年5月23日

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H. G. Nagaraja,
Pavithra R. C.

摘要

This study investigates the geometric properties of almost $\varvec{\eta }$ $η$ -Ricci-Bourguignon solitons on para-Kenmotsu manifolds. First, it is shown that if the metric $\varvec{g}$ $g$ on a para-Kenmotsu manifold admits an almost $\varvec{\eta }$ $η$ -Ricci-Bourguignon soliton and the Reeb vector field $\varvec{\xi }$ $ξ$ preserves the function $\varvec{\mu }$ $μ$ , then the manifold $\varvec{M}$ $M$ is Einstein. Next, we establish that if $\varvec{g}$ $g$ is an almost $\varvec{\eta }$ $η$ -Ricci-Bourguignon soliton and the vector field $\varvec{\xi }$ $ξ$ leaves both $\varvec{\lambda + \rho r}$ $λ + ρ r$ and $\varvec{\mu }$ $μ$ invariant, then the soliton reduces to an $\varvec{\eta }$ $η$ -Ricci-Bourguignon soliton. Finally, we prove that if $\varvec{g}$ $g$ is a gradient almost $\varvec{\eta }$ $η$ -Ricci-Bourguignon soliton, then $\varvec{M}$ $M$ is necessarily Einstein.

GEOMETRIC CHARACTERIZATIONS OF ALMOST \(\eta \)-RICCI-BOURGUIGNON SOLITONS ON PARA-KENMOTSU MANIFOLDS

摘要