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.

Edit: In light of the confusion my wording has caused I should make the following clarification: By {\it non-symmetric metric} I mean a collection of non-degenerate, bilinear maps for each tangent spaces $g_p:T_p(M) \times T_p(M) \to R$ for $p \in M$, inducing a map from $\chi(M) \times \chi(M) \to C^{\infty}(M)$ ($\chi(M)$ being the vector fields of $M$).

So by {\it non-symmetry} I mean that we are removing the requirement that $$ g_p(v,w) = g_p(w,v). $$