In this paper, we generalize the Cheng's maximal diameter theorem and Bishop volume comparison theorem to the manifold with the Bakry-Emery Ricci curvature. As their applications, we obtain some rigidity theorems on the warped product.
"Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature." Proc. Japan Acad. Ser. A Math. Sci. 85 (6) 71 - 74, June 2009. https://doi.org/10.3792/pjaa.85.71