In this paper, we study generalized Berwald square metrics <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F=\alpha +2\beta +\beta ^2/\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mn>2</mn> <mi>β</mi> <mo>+</mo> <msup> <mi>β</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha =\sqrt{a_{ij}(x)y^iy^j}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <msqrt> <mrow> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>y</mi> <mi>i</mi> </msup> <msup> <mi>y</mi> <mi>j</mi> </msup> </mrow> </msqrt> </mrow> </math></EquationSource> </InlineEquation> is a Riemannian metric and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta =b_i(x)y^i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>y</mi> <mi>i</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a one-form on a manifold M. Let F be a generalized Berwald square metric with isotropic S-curvature. We show that F is a generalized Douglas–Weyl metric if and only if it is R-quadratic if and only if it is R-reversible. We also prove that F is Ricci-quadratic if and only if it is Ricci-reversible. Finally, we show that every weakly Einstein square metric is Ricci-reversible if and only if it is Ricci-quadratic.

On Generalized Berwald Square Metrics with Isotropic S-Curvature

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发表日期：

2026年5月22日

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Sina Hedayatian

摘要

In this paper, we study generalized Berwald square metrics $F=\alpha +2\beta +\beta ^2/\alpha $ $F = α + 2 β + β^{2} / α$ where $\alpha =\sqrt{a_{ij}(x)y^iy^j}$ $α = \sqrt{a_{ij} (x) y^{i} y^{j}}$ is a Riemannian metric and $\beta =b_i(x)y^i$ $β = b_{i} (x) y^{i}$ is a one-form on a manifold M. Let F be a generalized Berwald square metric with isotropic S-curvature. We show that F is a generalized Douglas–Weyl metric if and only if it is R-quadratic if and only if it is R-reversible. We also prove that F is Ricci-quadratic if and only if it is Ricci-reversible. Finally, we show that every weakly Einstein square metric is Ricci-reversible if and only if it is Ricci-quadratic.