Weighted Ricci curvature in Riemann-Finsler geometry

Document Type : Original Article

Author

Department of Mathematical Sciences Indiana University-Purdue University 402 N Blackford Street Indianapolis, IN 46202 USA

10.22060/ajmc.2021.20473.1067

Abstract

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.

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