Consider the manifold $\mathbb{R^2}\setminus \{0\}$, on which the group of rotation acts. The orbits of the group are the circles centered in the origin, and form a foliation of $\mathbb{R^2}\setminus \{0\}$. This foliation will be denoted by $F\_1$. The foliation $F\_1$ defines univocally another foliation $F\_2$, with the following property: the tangent spaces of two leafs of $F\_1$ and $F\_2$ are orthogonal at the intersection point. In this case $F\_2$ is composed by the radial lines from the origin.


My question is the following: to what extent this situation can be generalized, i.e.: assume to have a riemanian manifold $M$, possibly flat, with a foliation $F\_1$ defined by the orbits of a group acting on $M$. To what extent does this foliation define univocally an orthogonal foliation $F\_2$ with the property that the tangent spaces of any pair of leafs of $F\_1$ and $F\_2$ are orthogonal at the intersection point(s)?



Orthogonal foliations 
There's a distinction to be made between two notions: *foliations* and *distributions*.


A **distribution** is the data, at each point *m* of *M*, of a subspace of *Tm*(*M*). These subspaces are all of the same dimension (say *r*), and depend smoothly on the point *m*, which means that they are generated by *r* smooth vector fields.


A **foliation** is a partition of the manifold into (not necessarily closed) submanifolds, such that, locally, this partition looks like the standard decomposition of ℝ*n* into translates of ℝ*d*. Ok, there's a caveat in my description since a same leaf could come infinitely many often in the neighborhood of a given point *m*. Anyways... I'm assuming that you know what a foliation is.



> 
> Foliations of *M* form a subset of distributions on *M*.
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> 
> The **Frobenius integrability criterion** (mentioned by Tom in him remark) states that a distribution *D* comes from a foliation iff for any vector fields *v* and *w* tangent to *D*, their [Lie bracket](http://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields) is again tangent to *D*.
> 
> 
> It turns out that that criterion is *always* satisfied for one-dimensional distributions, and so one-dimensional distributions are indeed in bijection with one-dimensional foliations. But that's no longer true for *r* ≥ 2.
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> 
> 


The operation of taking orthogonal complement is a very good operation for distributions: it's always well defined, and the orthogonal complement of the orthogonal complement is the distribution you started with.


But the orthogonal complement of a foliation is typically only a distribution.
The standard example that illustrates that situation is the vector field sin(*z*)*d/dx* + cos(*z*)*d/dy* on ℝ3. It defines a perfectly good foliation, but its orthogonal fails to satisfy the Frobenius integrability criterion, and therefore fails to be a foliation (in this particular case, it's a [contact structure](http://en.wikipedia.org/wiki/Contact_geometry), another beautiful mathematical notion...).




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Ah! You also wanted the foliation to be defined by the orbits of a group acting by isometries... That can be arranged: take the action of *S*1 on *S*3 given by the Hopf fibration. The orthogonal distribution is the standard contact structure on *S*3.


You also said that you wanted you Riemanninan manifold to be flat... In that case, you can take ℝ4=ℂ2 with its *S*1-action by complex multiplication. That example contains the above *S*3 example as an 
invariant submanifold, and therefore reproduces all its features.


