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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    R   >  0    ‐action. There are many examples admitting this structure. For example, (a) a class of pseudo‐Hessian manifolds admitting a free    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    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  ∘  f  )  g  )   , where    v  :   R   >  0    →   R   >  0is 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) with respect to    (  v  ∘  f  )  g    in terms of those on the level set of   f  . In particular, (1) we can generalize the result of Clarke and Rubinstein [ Ann. Inst. H. Poincaré Anal. NonLinéaire  30 (2013) 251‐274] about the metric completion of the space of Riemannian metrics with respect to the conformal transformations of the Ebin metric, and (2) two canonical Riemannian metrics on the    G  2    moduli space have different metric completions.

Conformal transformations of the pseudo‐Riemannian metric of a homogeneous pair