Weighted Ricci curvature in Riemann-Finsler geometry

Ricci curvature is one of the important geometric quantities in Riemann-Finsler geometry. Together with the S-curvature, one can dene 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 elds. 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