In the article http://www.sciencedirect.com/science/article/pii/0370269379905896 authors consider principal bundle $P(M, G)$ and then define induced metric $g$ on $\eta = Sp(A)/G$, where $Sp(A)$ - space of all gauge connections, $G$ - Lie group. They define $g(A, B) = (\tau_A, \tau_B)$, where $A, B$ - two vectors tangent to $\eta$ at the point $a \in \eta$, $\tau_A, \tau_B$ - their horizontal lifts respectively.
Then authors obtain expression for induced metric: $g(A, G) = (u_A, \Pi_\omega u_B) = (u_A, \Pi_{\omega_0} \Pi_\omega \Pi_{\omega_0} u_B)$, where $\Pi_\omega$ is projection operator onto horizontal subspace of $Sp(\omega)$ (here gauge connection $A$ was redefined as $\omega$), $\omega_0$ is some reference (background) gauge, $u_A, u_B$ - components of tangent (to $\eta$) vectors $A, B$ in $Sp(\omega)$.
And then they say that to calculate $\det g$ $g$ may be considered as a mapping from $S_0$ to $S_0$, where $S_0$ - connections satisfying reference gauge condition. They represent metric $g$ as
$g = 1 - \Pi_{\omega_0} \nabla_\omega G_\omega \nabla^{*}_\omega \Pi_{\omega_0}$,
where $\nabla$ is covariant derivative, $G_\omega = \frac{1}{\nabla^{*}_\omega \nabla_\omega}$.
Here I wonder, how we can put operator in correspondence to metric. Is there any rule?
And also I wonder, why we can use associate induced metric with scalar product of horizontal lifts?