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 k with geodesic flow, if an integral curve of k is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if k is complete, hypersurface-orthogonal, and satisfies Ric(k,k)≥0, then its divergence must be nonnegative. As an application, we show that if the Riemannian 3-manifold is closed and a unit length k with geodesic flow satisfies Ric(k,k)>0, then k cannot be hypersurface-orthogonal, thus recovering a result in Harris and Paternain (2013). Turning next to scalar curvature, we derive an evolution equation for the scalar curvature in terms of unit vector fields k that satisfy the condition R(k,⋅,⋅,⋅)=0. When the scalar curvature is a nonzero constant, we show that a hypersurface-orthogonal unit vector field k satisfies R(k,⋅,⋅,⋅)=0 if and only if it is a Killing vector field.