I have trouble putting down the notion of a homogeneous Riemannian metric.
Suppose we are given a Riemannian manifold $(M,g)$ on which a compact Lie group $G$ acts transitively by isometries (this means that $h^*g = g$ for any $h \in G$, i.e. for any vector fields $X,Y$ on $M$ and any point $p \in M$, $$ g_{h \cdot p} \big(h_{*,p}(X_p),h_{*,p}(Y_p)\big) = g_p\big(X_p,Y_p) \,. $$ It also means that if we fix any point $p \in M$ and denote by $K$ the isotropy subgroup associated with $p$ then there is a smooth bijection $\varphi \colon G / K \to M$ given by $\varphi(hK) = h \cdot p$.
Ok, this map then allows us to endow $G / K$ with a metric $\tilde g$ as follows: we decompose the Lie algebra $\mathfrak g$ of $G$ into $\mathfrak m \oplus \mathfrak k$ where $\mathfrak k$ is the Lie subalgebra of $K$ in $\mathfrak g$. Then we have an identification $T_pM \cong \mathfrak m \cong T_K(G/K)$, which allows us to set the value of $\tilde g$ at the point $hK$ on vector fields $X,Y$ over $G/K$ by $$ \tilde g_{hK}\big((L_{hK})_{*}(X_{K}),(L_{hK})_{*}(Y_{K})\big) := g_{p}\big(X_{K},Y_{K}\big) $$ where $L_{hK}$ denotes left translation in $G / K$ by $hK$ and we use the isomorphism $T_pM \cong \mathfrak m \cong T_K(G/K)$ to identify $X_K$ and $Y_K$ with elements in $T_pM$. This metric is manifestly invariant under left translation by $G$ (simply because the right hand side does not see the effect of left translation), i.e. $\tilde g$ is a $G$ - invariant metric on $G / K$.
Is this also called a homogeneous metric? I know what homogeneous Riemannian manifolds are, these are precisely of the form given above, where a Lie group acts transitively by isometries, however when it comes to the notion of a homogeneous metric I would have to \emph{guess} that by this is meant the construction above .. is it correct? In other words, do we mean then that $(G/K,\tilde g)$ is a homogeneous Riemannian manifold? Thanks a lot for your feedback and help!!