A new geometrical definition of a naturally reductive Finsler manifold using a geodesic graph is proposed, along with a possible generalization. Based on a construction from a recent paper by the authors, Finsler metrics based on naturally reductive Riemannian metrics <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> are studied. Explicit examples of purely Finsler naturally reductive <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha _i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-type metrics are constructed. Geodesic graphs on broad classes of Finsler <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha _i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-type metrics F which are derived from naturally reductive Riemannian metrics and which are not naturally reductive are described. The influence of one-forms <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta _j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>β</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> on the structure of geodesics of the metric F is also demonstrated and an explicit construction of families of Finsler naturally reductive metrics of the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\alpha _i,\beta _j)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>β</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-type is described.

The Natural Reductivity in Finsler Geometry in Terms of Geodesic Graphs

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发表日期：

2025年12月7日

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Teresa Arias-Marco,
Zdeněk Dušek

摘要

A new geometrical definition of a naturally reductive Finsler manifold using a geodesic graph is proposed, along with a possible generalization. Based on a construction from a recent paper by the authors, Finsler metrics based on naturally reductive Riemannian metrics $g_i$ $g_{i}$ are studied. Explicit examples of purely Finsler naturally reductive $\alpha _i$ $α_{i}$ -type metrics are constructed. Geodesic graphs on broad classes of Finsler $\alpha _i$ $α_{i}$ -type metrics F which are derived from naturally reductive Riemannian metrics and which are not naturally reductive are described. The influence of one-forms $\beta _j$ $β_{j}$ on the structure of geodesics of the metric F is also demonstrated and an explicit construction of families of Finsler naturally reductive metrics of the $(\alpha _i,\beta _j)$ $(α_{i}, β_{j})$ -type is described.