On the completeness of gradient Ricci solitons

Abstract

A gradient Ricci soliton is a triple $(M,g,f)$ satisfying $R_{ij}+\nabla _i\nabla _j f =\lambda g_{ij}$ for some real number $\lambda$. In this paper, we will show that the completeness of the metric $g$ implies that of the vector field $\nabla f$.