On the Completeness of Gradient Ricci Solitons

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