Suppose we have a smooth (unit) vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$. I am interested in properties of the family of surfaces orthogonal to $v$; i.e. the set of all possible space curves orthogonal to $v$.
One source I have looked at it is Aminov, Yu. Geometry of Vector Fields, 2000, in which this construct is called the non-holonomic manifold. However, there is not as much detail about it as I would like, particularly geometric or even physical interpretations.
Does anybody know of other ways to characterize such surfaces? I am obviously interested in the surfaces generated by this construct from a differential geometric perspective, but since every point has an infinite number of such space curves going through it, I was wondering if there were variational or functional analytic ways of looking at it as well (i.e. analyzing it as a space of functions).