Curvature of a homogenous manifold.

Question

I was a reading a paper and it seemed to me that in one of the equations the authors used the fact if $M$ is a homogenous Riemannian manifold (i.e., the group of isometries of $M$ act transitively on $M$) then the scalar curvature of $M$ is constant.

Is this fact true? I was unable to prove this. Is there a simple proof (or not so simple proof) of this fact?

P.S. : The manifold was also a gradient Ricci soliton. Can that make the scalar curvature constant.

Thanks

Answer 1

Since the curvature tensor is constructed naturally from the metric, any isometry is compatible with the curvature tensor, and since scalar curvature is obtained from the curvature tensor by a natural operation, the same argument applies here. But for scalar curvature this compatibility simply means the $R\circ f=f$ for any isometry $f$. Since on a homogeneous Riemannian manifolds the isometries act transitively, $R$ has to be constant.