A Riemannian manifold with constant sectional curvature is Einstein.

Question

A Riemannian manifold with constant sectional curvature is Einstein. Why?

It's true the inverse?

Answer 1

By definition, a Riemannian manifold has constant sectional curvature if the sectional curvature $K$ is a constant that is independent of the point and $2$-plane chosen. If $R$ denotes the covariant curvature tensor and $g$ is the metric then, as a consequence of the definition of $K$, the components satisfy the relation $$R_{ljhk} = K(g_{lh}g_{jk}-g_{lk}g_{jh}).$$

Multiplication with the contravariant metric tensor $g^{jk}$ yields $$R_{lh} = K(ng_{lh} - g_{lh}) = K(n-1)g_{lh},$$

from which we conclude that our manifold is Einstein.

For a counterexample of the converse, note that $\mathbb{C}P^n$ is Einstein but its sectional curvature is not constant (except for the sphere $n=1$).

However I believe that the converse is true for manifolds of dimension $\leq 3$.