Product Einstein Manifolds

Question

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?

Answer 1

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.