Product Einstein Manifolds In the book Einstein Manifolds by Besse it states the product of two Riemannian manifolds which are Einstein with the same constant $\lambda$ is an Einstein manifold with the same constant $\lambda$. Can someone provide a proof of this? Also what happens if the two manifolds are Einstein with different constants. Is the resulting product manifold still Einstein?

The answer to all your questions can be derived from the fact that if $(M,g)$ and $(N,h)$ are (pseudo)Riemannian manifolds, and $(P,k)$ is their product, then the metric tensor satisfies $$ k = \begin{pmatrix} g & 0\\\ 0 & h\end{pmatrix} $$ and the Ricci curvature satisfies $$ \mathrm{Ric}[k] = \begin{pmatrix} \mathrm{Ric}[g] & 0 \\\ 0 & \mathrm{Ric}[h] \end{pmatrix}. $$

The formula for the metric tensor is the definition of the product manifold. The formula for Ricci curvature can be found via a direct computation which is done it most textbooks in Riemannian geometry.