A note on static spaces and related problems

TL;DR

This paper addresses classification and conjecture problems related to Bach flat static spaces and critical point equations, providing new integral identities and results, especially in three dimensions, to characterize conformal flatness.

Contribution

It solves the classification problem for Bach flat static spaces and proves Besse's conjecture about critical point equations, with new integral identities in 3D.

Findings

Bach flat static spaces are classified.

Besse's conjecture on critical point equations is proved.

Conformal flatness is derived from divergence conditions in 3D.

Abstract

We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of the full divergence of the Cotton tensor for static metrics and metrics satisfying the critical point equation.