<p>In this paper, we study locally conformally flat almost gradient Ricci solitons. By combining the almost Ricci soliton equation with the vanishing of the Weyl conformal curvature tensor, we prove that every complete connected locally conformally flat almost gradient Ricci soliton is locally isometric to a warped product of a type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(t_1, t_2)\times _{\xi }\mathfrak {D}^{m-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mo>×</mo> <mi>ξ</mi> </msub> <msup> <mrow> <mi mathvariant="fraktur">D</mi> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> of a real interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(t_1, t_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and a space form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathfrak {D}^{m-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="fraktur">D</mi> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> of constant sectional curvature. The proof relies on geometric properties of the potential function, particularly the fact that its gradient is an eigenvector of both the Ricci tensor and the Hessian operator. Our result extends several rigidity and classification theorems for locally conformally flat gradient Ricci solitons to the broader framework of almost Ricci solitons.</p>

On locally conformally flat almost gradient Ricci solitons

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发表日期：

2026年7月20日

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Ali H. Alkhaldi,
Kamran Khan,
Pooja Bansal,
Akram Ali

摘要

In this paper, we study locally conformally flat almost gradient Ricci solitons. By combining the almost Ricci soliton equation with the vanishing of the Weyl conformal curvature tensor, we prove that every complete connected locally conformally flat almost gradient Ricci soliton is locally isometric to a warped product of a type $(t_1, t_2)\times _{\xi }\mathfrak {D}^{m-1}$ $(t_{1}, t_{2}) \times_{ξ} D^{m - 1}$ of a real interval $(t_1, t_2)$ $(t_{1}, t_{2})$ and a space form $\mathfrak {D}^{m-1}$ $D^{m - 1}$ of constant sectional curvature. The proof relies on geometric properties of the potential function, particularly the fact that its gradient is an eigenvector of both the Ricci tensor and the Hessian operator. Our result extends several rigidity and classification theorems for locally conformally flat gradient Ricci solitons to the broader framework of almost Ricci solitons.