TITLE: Ricci Curvature on Grassmannian
QUESTION [8 upvotes]: Suppose $G_r(n)$ is the Grassmannian, which is the collection of all $r$ dimensional subspace in $\mathbb{R}^{n}$ equipped with the usual invariant metric. Let $Ricc(G_r(n))$ be the Ricci curvature tensor. What are the best known constants $0<c_{n,r}<C_{n,r}$ such that
$$c_{n,r}\leq Ricc(G_r(n))\leq C_{n,r}$$?
I can only find results about complex Grassmannian, but not for real Grassmannian.

REPLY [6 votes]: Both the real and complex Grassmannians are compact irreducible symmetric spaces, and therefore are Einstein with positive Einstein constant.  (Thus $c_{n,r}=C_{n,r}$, and this constant can be varied by rescaling).  A reference is Besse, Einstein manifolds, paragraph 7.75.