  We introduce a new notion of a homogeneous pair for a pseudo-Riemannian
metric $g$ and a positive function $f$ on a manifold $M$ admitting a free
$\mathbb{R}_{>0}$-action. There are many examples admitting this structure. For
example, (a) a class of pseudo-Hessian manifolds admitting a free
$\mathbb{R}_{>0}$-action and a homogeneous potential function such as the
moduli space of torsion-free $G_2$-structures, (b) the space of Riemannian
metrics on a compact manifold, and (c) many moduli spaces of geometric
structures such as torsion-free ${\rm Spin}(7)$-structures admit this
structure. Hence we provide the unified method for the study of these geometric
structures.
  We consider conformal transformations of the pseudo-Riemannian metric $g$ of
a homogeneous pair $(g, f)$. Showing that the pseudo-Riemannian manifold $(M,
(v \circ f) g)$, where $v: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ is a
smooth function, has the structure of a warped product, we study the geometric
structures such as the sectional curvature, geodesics and the metric completion
(if $g$ is positive definite) w.r.t. $(v \circ f) g$ in terms of those on the
level set of $f$. In particular, (1) we can generalize the result of Clarke and
Rubinstein about the metric completion of the space of Riemannian metrics
w.r.t. the conformal transformations of the Ebin metric, and (2) two canonical
Riemannian metrics on the $G_2$ moduli space have different metric completions.
