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

Abstract

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.