1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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 $S^{2n+1}\rightarrow \mathbb{C}P^{n}$.

https://en.wikipedia.org/wiki/Hopf_fibration#Generalizations

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.