Let (B, $check{g}$) and (N, $hat{g}$) be Einstein manifolds. Then, we get a complete (necessary and sufficient) condition for the warped product manifold $B;{ imes}_f;N;:=;(B;{ imes};N,;check{g};+;f{hat{g}}$) to be Einstein, and obtain a complete condition for the Einstein warped product manifold $B;{ imes}_f;N$ to be weakly stable. Moreover, we get a complete condition for the map i : ($B,;check{g});{ imes};(N,;hat{g});{ ightarrow};B;{ imes}_f;N$, which is the identity map as a map, to be harmonic. Under the assumption that i is harmonic, we obtain a complete condition for $B;{ imes}_f;N$ to be Einstein.