Suppose $S$ is an orientable regular surface in $\mathbb R^3$ without umbilical points (not necessarily compact, and with no boundary). There are two well-defined smooth $1$-dimensional tangent distributions on $S$ corresponding to the directions of curvature.

> Can each of these be generated by a single unitary vector field?

Of course this is possible locally and a standard argument using Čech cocycles gets one from there to simply connected surfaces, which is not very much. This argument is generic —does not use the fact that these are the directions of curvature of the surface— so this is to be expected. It may well be the case that the cocycles one gets from this are special in some way which I cannot see, so that one does not need the vanishing of $\pi_1$.