The Newman-Penrose Formalism for Riemannian 3-manifolds

We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field ${\boldsymbol k}$ with geodesic flow, if an integral curve of ${\boldsymbol k}$ is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if ${\boldsymbol k}$ is complete, hypersurface-orthogonal, and satisfies $\text{Ric}({\boldsymbol k},{\boldsymbol k}) \geq 0$, then its divergence must be nonnegative. As an application, we show that if the Riemannian 3-manifold is closed and a unit length ${\boldsymbol k}$ with geodesic flow satisfies $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, then ${\boldsymbol k}$ cannot be hypersurface-orthogonal, thus recovering a recent result. Turning next to scalar curvature, we derive an evolution equation for the scalar curvature in terms of unit vector fields ${\boldsymbol k}$ that satisfy the condition $R({\boldsymbol k},\cdot,\cdot,\cdot) = 0$. When the scalar curvature is a nonzero constant, we show that a hypersurface-orthogonal unit vector field ${\boldsymbol k}$ satisfies $R({\boldsymbol k},\cdot,\cdot,\cdot) = 0$ if and only if it is a Killing vector field.