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<p>Ricci solitons represent one of the most natural generalizations of Einstein metrics, being objects of great interest in some theories of modern physics. In turn, almost Ricci solitons - a generalization of Ricci solitons obtained by substituting the soliton constant with a smooth function, have become a fervent topic in the last decade mainly due to their important applications in theoretical physics and geometry. The aim of this manuscript is to investigate almost Ricci solitons and gradient Ricci solitons on Riemannian concircular structure 3-manifolds (briefly, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((RCS)_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>C</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-manifolds). We first deduce the conditions under which an almost Ricci soliton on an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((RCS)_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>C</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-manifold is steady, expanding, or shrinking. Then we prove that the scalar curvature of an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((RCS)_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>C</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-manifold endowed with an almost Ricci soliton stands as a non-trivial solution for two PDE’s of high interest in medical science, physics and engineering, namely the screened Poisson equation and the inhomogeneous Helmholtz equation. Finally, we construct a meaningful example of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((RCS)_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>C</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-manifold with a gradient Ricci soliton, illustrating some of our results.</p>

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Almost Ricci solitons on Riemannian concircular structure 3-manifolds

  • 期刊论文

  • 发表日期：

    2025年12月27日

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  • Bang-Yen Chen,
  • Sudhakar Kumar Chaubey,
  • Uday Chand De,
  • Gabriel-Eduard Vîlcu

摘要

Ricci solitons represent one of the most natural generalizations of Einstein metrics, being objects of great interest in some theories of modern physics. In turn, almost Ricci solitons - a generalization of Ricci solitons obtained by substituting the soliton constant with a smooth function, have become a fervent topic in the last decade mainly due to their important applications in theoretical physics and geometry. The aim of this manuscript is to investigate almost Ricci solitons and gradient Ricci solitons on Riemannian concircular structure 3-manifolds (briefly, \((RCS)_3\) ( R C S ) 3 -manifolds). We first deduce the conditions under which an almost Ricci soliton on an \((RCS)_3\) ( R C S ) 3 -manifold is steady, expanding, or shrinking. Then we prove that the scalar curvature of an \((RCS)_3\) ( R C S ) 3 -manifold endowed with an almost Ricci soliton stands as a non-trivial solution for two PDE’s of high interest in medical science, physics and engineering, namely the screened Poisson equation and the inhomogeneous Helmholtz equation. Finally, we construct a meaningful example of \((RCS)_3\) ( R C S ) 3 -manifold with a gradient Ricci soliton, illustrating some of our results.