Uniqueness of static decompositions

TL;DR

This paper classifies static manifolds with multiple static decompositions under curvature conditions, revealing that the base manifold is static and the warping function is highly restricted, ensuring uniqueness in certain Lorentzian cases.

Contribution

It provides a classification of static manifolds with multiple decompositions and establishes conditions for uniqueness in Lorentzian manifolds.

Findings

The base manifold is itself static.

Warping functions satisfy severe restrictions.

Certain lightlike curvature conditions ensure uniqueness.

Abstract

We classify static manifolds which admit more than one static decomposition whenever a condition on the curvature is fullfilled. For this, we take a standard static vector field and analyze its associated one parameter family of projections onto the base. We show that the base itself is a static manifold and the warping function satisfies severe restrictions, leading us to our classification results. Moreover, we show that certain condition on the lightlike sectional curvature ensures the uniqueness of static decomposition for Lorentzian manifolds.