  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.
