Compact manifolds with positive m-Bakry-Émery Ricci tensor

In this paper we prove some rigidity theorems for complete manifold N with Rcfm≥(m−1)c>0 by the existence of a nice f-minimal hypersurface, this may be regard as a Myers-Cheng type theorem replaced Ricci curvature by m-Bakry-Émery Ricci curvature. First, we prove an upper bound for the distance function to a nice f-minimal hypersurface embedded in N. We then consider the rigidity part when the upper bound is achieved, we also give a first nonzero Dirichlet eigenvalue comparison for the f-Laplacian on manifold with m-Bakry-Émery Ricci curvature bounded from below.