In this paper, we study a class of Finslerian almost Ricci solitons, called almost square Ricci solitons, defined by a square metric <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F=(\alpha +\beta )^2/\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> and a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> vector field V on an n-dimensional manifold M, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> are respectively a Riemannian metric and a 1-form on a manifold M. We prove that (M, F, V) is an almost square Ricci soliton if and only if F is Ricci flat and V is a conformal vector field of F when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and it is a locally projectively flat almost square Ricci soliton if and only if F is of zero flag curvature and V is a Killing vector field of F when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. As applications, we determine the structures of (locally projectively flat) almost square Ricci solitons.

On almost square Ricci solitons

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发表日期：

2025年11月15日

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Mingzhu Wang,
Qiaoling Xia

摘要

In this paper, we study a class of Finslerian almost Ricci solitons, called almost square Ricci solitons, defined by a square metric $F=(\alpha +\beta )^2/\alpha $ $F = {(α + β)}^{2} / α$ and a $C^2$ $C^{2}$ vector field V on an n-dimensional manifold M, where $\alpha $ $α$ and $\beta $ $β$ are respectively a Riemannian metric and a 1-form on a manifold M. We prove that (M, F, V) is an almost square Ricci soliton if and only if F is Ricci flat and V is a conformal vector field of F when $n\ge 2$ $n \geq 2$ , and it is a locally projectively flat almost square Ricci soliton if and only if F is of zero flag curvature and V is a Killing vector field of F when $n\ge 3$ $n \geq 3$ . As applications, we determine the structures of (locally projectively flat) almost square Ricci solitons.