Einstein manifold

Question

Does an Einstein manifold have to have integer $\lambda$ constant?

Why a Riemannian product between two Einstein manifoldd with different $\lambda$ constant is not Einstein?

A manifold with constant sectional curvature is Einstein, while a manifold with constant scalar curvature may not be Einstein, why?

Answer 1

• $\lambda$ can be any real number. It does not have to be an integer.
• If $M=M_1\times M_2$ with $Ric_i=\lambda_ig_i$ then $Ric(X_1,X_1)=\lambda_1 g_1(X_1,X_1)$ and $Ric(X_2,X_2)=\lambda_2 g_2(X_2,X_2)$. Thus if $\lambda_1\ne \lambda_2$ then $M$ is not Einstein.
• Having constant sectional curvature is more restrictive than having constant Ricci curvature. And having constant Ricci curvature is more restrictive than having constant scalar curvature. Thus: constant sectional curvature implies constant Ricci curvature (that is, Einstein) implies constant scalar curvature. But the converse converse implications are not true.