2021/11/30 by Kuwae, Kazuhiro, Shukuri, Toshiki
#53C20 #53C22 #53C23 #53C24 #58J60 #Differential Geometry (math.DG) #FOS: Mathematics #Secondary 53C21
paper · doi:10.48550/arxiv.2111.15508
In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth n-dimensional Riemannian manifold having a lower bound of modified m-Bakry-Émery Ricci tensor under m≤ 1 in terms of vector fields. As consequences, we give the optimal conditions for modified m-Bakry-Émery Ricci tensor under m≤1 such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for m-Bakry-Émery Ricci curvature under m≥ n if the vector field is a gradient type. When m<1, our results are new in the literature.