Electrostatic system with divergence-free Bach tensor and non-null cosmological constant

TL;DR

This paper proves that three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat under certain conditions, leading to a warped product structure with specific geometric properties.

Contribution

It establishes a new geometric characterization of electrostatic manifolds with divergence-free Bach tensor in three dimensions, linking conformal flatness to the dependence of electric field and lapse function gradient.

Findings

Manifolds are locally conformally flat when electric field and lapse gradient are dependent.

Such manifolds admit a warped product structure with constant curvature fibers.

The results connect divergence-free Bach tensor condition to geometric and conformal properties.

Abstract

We prove that three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat, provide that the electric field and the gradient of the lapse function are linearly dependent. Consequently, a three-dimensional electrostatic manifold admits a local warped product structure with a one-dimensional base and a constant curvature surface fiber.