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

## TL;DR

This paper introduces a new framework for analyzing pseudo-Riemannian metrics on manifolds with a free positive action, unifying various geometric structures and studying their conformal transformations and metric completions.

## Contribution

It defines homogeneous pairs for pseudo-Riemannian metrics and positive functions, providing a unified approach to study conformal transformations and geometric properties of related moduli spaces.

## Key findings

- The conformal transformation $(v \\circ f) g$ forms a warped product structure.
- Generalizes metric completion results for Riemannian metric spaces.
- Shows different metric completions for canonical metrics on $G_2$ moduli space.

## 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.