A theorem of Ambrose for Bakry–Emery Ricci tensor

In this short note, we prove a theorem of Ambrose (or Myers) for the Bakry–Emery Ricci tensor with the potential function at most linear growth. We also prove a complete manifold (M,g,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M, g, f)$$\end{document} with the Bakry–Emery Ricci tensor bounded from below by a uniform positive constant and the potential function at most quadratic growth is compact.