Three-Dimensional Almost Contact Metric Manifolds Revisited via the Newman-Penrose Formalism

TL;DR

This paper adapts the Newman-Penrose formalism from General Relativity to analyze three-dimensional almost contact metric manifolds, enabling new classification results for certain compact manifolds with special metrics.

Contribution

It introduces a novel application of the Newman-Penrose formalism to ACM manifolds and classifies compact normal ACM manifolds admitting an η-Einstein metric.

Findings

Reformulation of ACM manifold properties using Newman-Penrose formalism

Classification of compact normal ACM manifolds with η-Einstein metrics

Demonstration of the formalism's utility in geometric analysis

Abstract

This paper applies the Newman-Penrose formalism-a technique primarily used in General Relativity-to the analysis of three-dimensional almost contact metric (ACM) manifolds. We reformulate and discuss several known notions and properties within the Newman-Penrose framework, demonstrating the applicability of the method in this geometric context. Furthermore, as an application showcasing the utility of the formalism, we address the classification of three-dimensional compact normal ACM manifolds, or equivalently trans-Sasakian manifolds, that admit an $\eta$-Einstein metric.