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

Abstract

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.