For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric condition is relaxed, do there exist a large number of non-symmetric equivariant Riemannian metrics. Also, what is a specific example of a non-symmetric Riemannian metric on complex projective $n$-space, I am having difficulty coming up with one.