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?



Complementary integrable vector fields
The obstruction against integrability of the orthogonal is called curvature
for the (Ehresmann) connection given by orthogonal projection onto the distribution generated by $X$. See section 17 of 


* Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008. [(pdf)](http://www.mat.univie.ac.at/~michor/dgbook.pdf)


where the situation is slightly more special (the orbit space is a manifold, and the projection is a fiber bundle).
Or see the paper for a more general situation (the vertical bundle, here generated by $X$, need not be integrable either, and then you have curvature and cocurvature):


* Peter W. Michor: Graded derivations of the algebra of differential forms associated with a connection. Differential Geometry, Pe\ niscola, 1988, \eds F.J. Carreras, O. Gil-Medrano, A.M. Naveira Lecture Notes in Math. 1410 (1989), 249--261, Springer-Verlag, Berlin. [(pdf)](http://www.mat.univie.ac.at/~michor/der.pdf)


In the case of $S^3$ and the Hopf fibration, the orbit space is $S^2$, and the connection is a principal $S^1$-connection whose curvature 2-form is a multiple of the volume form on $S^2$, if I remember correctly. 


