Suppose I have $\mathbb{S}^5$, foliated by Hopf circles. I am wondering if this restricts in some way to the foliation by Hopf circles on $\mathbb{S}^3$ in the join $\mathbb{S}^5=\mathbb{S}^3*\mathbb{S}^1$.
More generally, given a foliation of $\mathbb{S}^{2n+1}$ by Hopf circles, does this foliation restrict to Hopf circle foliations of $\mathbb{S}^{2k+1}$ in the join $\mathbb{S}^{2n+1}=\mathbb{S}^{2k+1}*\mathbb{S}^{2(n-k)-1}$?
I think the answer is yes. If we think of $\mathbb{S}^5$ as the units in $\mathbb{C}^3$, then the foliation by Hopf circles is induced by the action of $\mathbb{S}^1$ described by $z\cdot (z_1,z_2,z_3)=(zz_1,zz_2, zz_3)$ for $z\in \mathbb{S}^1$.
Now the set of points in $\mathbb{S}^5$ with $z_3=0$ is an $\mathbb{S}^3=\{(z_1,z_2,0): |z_1|^2+|z_2|^2=1\}$. The orbits under the action not only stay in this $\mathbb{S}^3$, but are precisely the circles given by the Hopf action of $\mathbb{S}^1$ on $\mathbb{S}^3$. Moreover, this particular $\mathbb{S^3}\subset \mathbb{S}^5$ is "dual" to $\mathbb{S}^1=\{(0,0,z_3):|z_3|^2=1\}$. So in the join $\mathbb{S}^5=\mathbb{S}^3*\mathbb{S}^1$ (of these particular subspheres), we see that the Hopf foliation on $\mathbb{S}^5$ restricts to the Hopf foliation on $\mathbb{S}^3$.
This should hold in general as well.