I am having a question on how to induce a metric $g$ on homogeneous space $G/H$, if one is given a ${\rm Ad}_H$-invariant metric $\bar{g}$ on G.
More specifically and simply, consider principal bundle $\pi: G \rightarrow G/H$, where $G$ is compact and simple Lie group and $H$ is its closed subgroup. We have the Killing form $K(X, Y)={\rm Tr}({\rm ad_X}\ {\rm ad_Y})$ to define a metric $\bar{g}$ on G, where $X, Y\in \mathfrak{g}$. Apparently metric $\bar{g}$ is ${\rm Ad}_H$-invariant.
Now if one decompose $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$, one can find an isomorphism between $G\times_{{\rm Ad}_H}\mathfrak{m}$ and $T(G/H)$, or an isomorphism of $T_o(G/H)$ and $\mathfrak{m}$. Further a left $k$-action $L_k$on $G$ induces a isometric transformation $\tau_k: gH \mapsto kgH$.
My question is that
$\bullet$ I am wondering if I am right that for every decomposition choice of $\mathfrak{m}$, one can define the metric $g$ on $G/H$, such that $g_{kH}=\tau_{k^{-1}}^\ast\circ K\vert_{\mathfrak{m}}$, and $g$ is $G$-invariant metric on $G/H$
$\bullet$ if the above is correct, is the Levi-Civita connection $\Gamma$, due to the metric $g$ defined above, just the connection by choosing the horizontal space $H_k=L_{k\ast}\mathfrak{m}$ on $T_kG$?
Thanks in advance!