Let $(X^{m+n},g)\equiv(\mathbb{R}^{m}\times K^{n},g_{\mathbb{R}^{m}}\oplus h)$ be a Riemann product of a Euclidean space $(\mathbb{R}^{m},g_{\mathbb{R}^{m}})$ and a compact Riemannian manifold $(K^{n},h)$. For any point $p=(p_{1},p_{2})\in X^{m+n}$, there exists a Green’s function $G_{p}$ on $X$ such that

1. (1)

   $-\Delta_{g}G_{p}=2\pi\delta_{p}$.

2. (2)

   There are constants $\epsilon>0$, $R>0$ and $C>0$, independent of $p$, such that

   (8.1)

   $$|G_{p}(x)-\Phi_{m}(x_{1})|\leq Ce^{-\epsilon\cdot|x_{1}-p_{1}|}$$

   for any $x=(x_{1},x_{2})\in X^{m+n}\setminus B_{R}(p)$, where $\Phi_{m}:\mathbb{R}_{+}\to\mathbb{R}$ is the standard Green’s function on $\mathbb{R}^{m}$ with a singularity at $p_{1}$.