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<p>The Ricci version of the Schur theorem is shown to hold for a wide class of Finsler metrics. What is more, let <i>F</i> be any Finsler metric whose Ricci curvature is a function <i>ρ</i>: <i>M</i><sup><i>n</i></sup> → ℝ (i.e., (<i>M</i><sup><i>n</i></sup>, <i>F</i>) is Einstein), where <i>n</i> ≥ 3. For <i>x</i> ∈ <i>M</i>, we express d<i>ρ</i><sub><i>x</i></sub> as an average over the indicatrix in T<sub><i>x</i></sub><i>M</i> of the Hilbert form weighted by a combination of derivatives of the Landsberg tensor. As a consequence of this general expression, if the metric is weakly Landsberg, then <i>ρ</i> must be constant. The proof is based on the invariance of certain functionals under Diff(<i>M</i>).</p><p>Furthermore, the Schur theorem holds for the class of pseudo-Finsler metrics with quadratic Ricci scalar. We extend previous arguments in the literature that proved this in particular cases. Lastly, we analyze the status of the general problem.</p>

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Schur theorem for the Ricci curvature of any weakly Landsberg Finsler metric

  • 期刊论文

  • 发表日期：

    2025年7月30日

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  • Fidel F. Villaseñor

摘要

The Ricci version of the Schur theorem is shown to hold for a wide class of Finsler metrics. What is more, let F be any Finsler metric whose Ricci curvature is a function ρ: Mn → ℝ (i.e., (Mn, F) is Einstein), where n ≥ 3. For x ∈ M, we express dρx as an average over the indicatrix in TxM of the Hilbert form weighted by a combination of derivatives of the Landsberg tensor. As a consequence of this general expression, if the metric is weakly Landsberg, then ρ must be constant. The proof is based on the invariance of certain functionals under Diff(M).

Furthermore, the Schur theorem holds for the class of pseudo-Finsler metrics with quadratic Ricci scalar. We extend previous arguments in the literature that proved this in particular cases. Lastly, we analyze the status of the general problem.