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