# A foliation of the sphere

In the context of coisotropic reduction, I asked myself the following question:

In $\mathbb{R}^{2n}$ with coordinates $(x_{1},y_{1},\ldots,x_{n},y_{n})$, we consider the unit sphere $S^{2n-1}\subset\mathbb{R}^{2n}$. There is a distribution on the sphere, generated by the vector field $$x_{1}\frac{\partial}{\partial y_{1}}-y_{1}\frac{\partial}{\partial x_{1}}+\cdots+x_{n}\frac{\partial}{\partial y_{n}}-y_{n}\frac{\partial}{\partial x_{n}}.$$

This gives rise to a foliation on $S^{2n-1}$. What does the leaf space $S^{2n-1}/\sim$ look like?

In case $n=1$, then the leaf space seems to be just one point. But what happens in the higher dimensional cases?

This a generalisation of the Hopf fibration. If you see $S^{2n+1}\subset \mathbb{C}^{n+1}$ the orbit of the low of this vector field are the fibres of $S^{2n+1}\rightarrow \mathbb{C}P^{n}$.