Some remarks on the generalized Myer’s theorems

Abstract

In this paper, firstly, we prove a generalization of Ambrose (or Myers) theorem for the Bakry-Emery Ricci tensor. Later, we improve previous diameter estimate obtained by Galloway for complete Riemannian manifolds. To obtain these results, we utilize the Riccati inequality and the index form of a minimizing unit speed geodesic segment, respectively.