Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric conditions we are sure that the codimension one distribution on $M$ orthogonal to $X$ (orthogonal to $F$) is integrable? Is there a global geometro-dynamical meaning for such possible conditions?

In particular, what is the answer for the standard metric of $S^{3}$ and its one dimensional foliation by circle arising from Hopf fibration?

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric conditions we are sure that the codimension one distribution on $M$ orthogonal to $X$ (orthogonal to $F$) is integrable? Is there a global geometro-dynamical meaning for such possible conditions?

In particular, what is the answer for the standard metric of $S^{3}$ and its one dimensional foliation by circle arising from Hopf fibration? In this particular case, if this 2 dim. distribution is integrable, to what extent this 2. dim foliation is studied?