Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 2 2 2021 09 01 Weighted Ricci curvature in Riemann-Finsler geometry 117 136 4500 10.22060/ajmc.2021.20473.1067 EN Zhongmin Shen Department of Mathematical Sciences, Indiana University-Purdue University, 402 N Blackford Street, Indianapolis, IN 46202, USA 0000-0001-8074-3944 Journal Article 2021 08 26 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. https://ajmc.aut.ac.ir/article_4500_7d6548bdc0082aacc950ed35e91fcccb.pdf