Killing vector fields on Riemannian and Lorentzian 3‐manifolds

We give a complete local classification of all Riemannian 3‐manifolds (M,g)$(M,g)$ 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 Ric(T,T)$\text{Ric}(T,T)$, where 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 Ric(T,T)=2$\text{Ric}(T,T) = 2$. We also give necessary, and separately, sufficient conditions, both expressed in terms of Ric(T,T)$\text{Ric}(T,T)$, 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 R3$\mathbb {R}^3$, 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.