Schur theorem for the Ricci curvature of any weakly Landsberg Finsler metric
| Autor: | Villaseñor, Fidel F. |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 (positive definite) Finsler metric such that $\text{Ric} =\rho F^2$ with $\rho\colon M^n\rightarrow\mathbb{R}$ (i.e., $(M^n,F)$ is Einstein) and $n\geq 3$. For $x\in M$, we express $\text{d}\rho_x$ as an average over the indicatrix in $\text{T}_xM$ of the Hilbert $1$-form weighted by a combination of derivatives of the mean Landsberg tensor. As a consequence of this general expression, if the metric is weakly Landsberg, then $\rho$ must be constant. The proof is based on the invariance of natural functionals under $\text{Diff}(M)$. Furthermore, we revisit an independent argument which proves the Schur theorem for the class of pseudo-Finsler metrics with quadratic Ricci scalar, improving previous results on the topic. Comment: 18 pages, no figures. The framework and proofs are built up in detail |
| Databáze: | arXiv |
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