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The Ricci version of the Schur theorem is shown to hold for a wide class ofFinsler metrics. What is more, let $F$ be any (positive definite) Finslermetric such that $\text{Ric} =\rho F^2$ with $\rho\colonM^n\rightarrow\mathbb{R}$ (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 derivativesof the mean Landsberg tensor. As a consequence of this general expression, ifthe metric is weakly Landsberg, then $\rho$ must be constant. The proof isbased on the invariance of natural functionals under $\text{Diff}(M)$.  Furthermore, we revisit an independent argument which proves the Schurtheorem for the class of pseudo-Finsler metrics with quadratic Ricci scalar,improving previous results on the topic.

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