Adapted metrics on locally conformally product manifolds

TL;DR

This paper demonstrates that the Gauduchon metric on certain compact manifolds is naturally adapted to the manifold's structure, and characterizes such metrics as critical points of a specific functional.

Contribution

It introduces the concept of adapted metrics on locally conformally product manifolds and characterizes them variationally.

Findings

Gauduchon metric is adapted with vanishing Lee form on the flat distribution

Adapted metrics are critical points of a natural conformal functional

Provides a new characterization of metrics on locally conformally product manifolds

Abstract

We show that the Gauduchon metric $g_0$ of a compact locally conformally product manifold $(M,c,D)$ of dimension greater than $2$ is adapted, in the sense that the Lee form of $D$ with respect to $g_0$ vanishes on the $D$-flat distribution of $M$. We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.