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

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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[1] D. Bao, S. S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, Springer-Verlag, 2000.
[2] X. Cheng and Z. Shen, Some inequalities on Finsler manifolds with weighted Ricci curvature bounded below, preprint, 2021.
[3] M. Gromov, Dimension, nonlinear spectra and width, Geometric aspects of fnctional analysis, Israel seminar (1986-87), Lecture Notes in Math., 1317, Springer, Berlin (1988), 132-184.
[4] M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. 95 (2010), 191-199.
[5] M. Limoncu, The Bakry-Emery Ricci tensor and its applications to some compactness, Math. Z. 271(2012), 715-722.
[6] S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Diff. Eq. 36 (2009), 211-249.
[7] S. Ohta and K. Sturm, Heat flow on Finsler manifolds, Comm. on Pure and Applied Math. 62(10) (2009), 1386-1433.
[8] Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser(2), 48 (1970), 235-242.
[9] Z. Shen, Lectures on Finsler geometry, World Scentific, 2001.
[10] S. Yin, Q. He and D. Zheng, Some comparison theorems and their applications in Finsler geometry, Journal of Inequalities and Applications, 2014(107) (2014).
[11] S. Yin, Two compactness theorems on Finsler manifolds with positivr weighted Ricci cirvature, Results Math, (2017). DOI 10.1007/s00025-017-0673-9 .
[12] S. Yin, Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below Frontiers of Mathematics in China 13(4) (2018), 1-14.
[13] G. Wei and W. Wylie, Comprison Geometry for the Bakry-Emery Ricci Tensor, J. Diff. Geom. 83 (2009), 377-405.