%0 Journal Article %T Weighted Ricci curvature in Riemann-Finsler geometry %J AUT Journal of Mathematics and Computing %I Amirkabir University of Technology %Z 2783-2449 %A Shen, Zhongmin %D 2021 %\ 09/01/2021 %V 2 %N 2 %P 117-136 %! Weighted Ricci curvature in Riemann-Finsler geometry %K Ricci curvature %K $S$-curvature %K Mean curvature %R 10.22060/ajmc.2021.20473.1067 %X Ricci curvature is one of the important geometric quantities in Riemann-Finsler geometry. Together with the $S$-curvature, one can define a weighted Ricci curvature for a pair of Finsler metric and a volume form on a manifold. One can build up a bridge from Riemannian geometry to Finsler geometry via geodesic fields. Then one can estimate the Laplacian of a distance function and the mean curvature of a metric sphere under a lower weighted Ricci curvature by applying the results in the Riemannian setting. These estimates also give rise to a volume comparison of Bishop-Gromov type for Finsler metric measure manifolds. %U https://ajmc.aut.ac.ir/article_4500_fc3d11fc673b4e9a44f3b4e11ecbea4e.pdf