  Inspired by Hongjie Dong and Qi S. Zhang's article \cite{ZQ2}, we find that
the analyticity in time for a smooth solution of the heat equation with
exponential quadratic growth in the space variable can be extended to any
complete noncompact Riemannian manifolds with Bakry-\'Emery Ricci curvature
bounded below and the potential function being of at most quadratic growth.
Therefore, our result holds on all gradient Ricci solitons. As a corollary, we
give a necessary and sufficient condition on the solvability of the backward
heat equation in a class of functions with the similar growth condition. In
addition, we also consider the solution in certain $L^p$ spaces with
$p\in[2,+\infty)$ and prove its analyticity with respect to time.
