Newman--Penrose formalism in $3$-dimensional trans-Sasakian manifolds

TL;DR

This paper applies the Newman--Penrose formalism to 3D trans-Sasakian manifolds, deriving curvature identities and rigidity results, and linking geometric structures to conformal foliations and harmonic morphisms.

Contribution

It introduces a scalar spin coefficient framework for 3D trans-Sasakian manifolds, connecting their geometry to conformal foliations and providing new curvature and rigidity results.

Findings

Trans-Sasakian condition is equivalent to shear-free geodesic congruence.

Derived curvature and Laplacian identities for main subclasses.

Proved rigidity: characteristic vector field is vertical in non-space-form cases.

Abstract

We study $3$-dimensional trans-Sasakian manifolds using the Newman--Penrose formalism. In this framework, the geometry of the structure vector field is encoded by scalar spin coefficients: acceleration, shear, expansion, and twist. A central observation is that, in dimension $3$, the trans-Sasakian condition is equivalent to the characteristic vector field defining a shear-free geodesic congruence, or equivalently a conformal foliation by geodesics. Thus, the Newman--Penrose equations provide a direct scalar formulation of the conformal foliations studied by Baird and Wood in the theory of harmonic morphisms. Within this framework, we derive curvature and Laplacian identities for trans-Sasakian manifolds and their main subclasses, including formulae for the Ricci tensor, scalar curvature, Einstein condition, rough Laplacian, divergence and harmonicity of the characteristic vector field, together with several illustrative examples. As an application, we consider trans-Sasakian structures compatible with fixed homogeneous metrics of type ${\Bbb E}(\kappa,\tau)$. We prove a rigidity result: in the non-space-form cases, the Newman--Penrose equations force the characteristic vector field to be vertical. Hence, for $\tau\neq0$ and $\kappa\neq4\tau^2$, every compatible trans-Sasakian structure is the canonical vertical $\alpha$-Sasakian structure, while for $\tau=0$ and $\kappa\neq0$, it is vertical and cosymplectic. In particular, these non-space-form homogeneous metrics admit no proper compatible trans-Sasakian structures.