Killing vector fields on Riemannian and Lorentzian 3-manifolds

TL;DR

This paper classifies all 3-dimensional Riemannian and Lorentzian manifolds with nonvanishing Killing vector fields, linking their geometry to scalar curvature and Ricci tensor components, and explores conditions for conformal flatness and completeness.

Contribution

It provides a complete local classification of 3-manifolds with Killing fields, extending Sasakian structures to Lorentzian cases, and establishes criteria for conformal flatness and geodesic completeness.

Findings

Classification of Riemannian 3-manifolds with Killing fields

Extension to Lorentzian 3-manifolds with timelike Killing fields

Conditions for conformal flatness and geodesic completeness

Abstract

We give a complete local classification of all Riemannian 3-manifolds $(M,g)$ admitting a nonvanishing Killing vector field $T$. We then extend this classification to timelike Killing vector fields on Lorentzian 3-manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature $S$ of $g$ and the function $\text{Ric}(T,T)$, where $\text{Ric}$ is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of $T$. Our classification generalizes that of Sasakian structures, which is the special case when $\text{Ric}(T,T) = 2$. We also give necessary, and separately, sufficient conditions, both expressed in terms of $\text{Ric}(T,T)$, for $g$ to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that $T$ has unit length and the coordinates derived in our classification are globally defined on $\mathbb{R}^3$, we give conditions under which $S$ completely determines when the metric will be geodesically complete. In the event that the 3-manifold $M$ is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.