# Periodic orbit for certain vector field on $S^3$ (à la Seifert conjecture)

The standard frame for $$S^3$$ consists of $$X_i,X_j,X_k$$ with $$X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$$ where $$i,j,k$$ are standard quaternion numbers, $$a\in S^3$$, and the multiplication is the quaternion multiplication. This global frame on $$S^3$$ gives us a trivialization of $$TS^3\simeq S^3\times \mathbb{R}^3$$. So the Hopf map $$P:S^3\to S^2\subset \mathbb{R}^3$$ is counted as a unit vector field on $$S^3$$. This vector field is called "Hopf vector field".

Does the Hopf vector field have a periodic orbit? Is this vector field discussed in various attempts to find an analytic counter example to Seifert conjecture?

• Isn't every orbit periodic because the Hopf fibration has fiber a circle? Commented Jun 21, 2019 at 23:36
• @DanRust Why do you think the fibers are invariant under flow of this vector foeld?Please see the construction of this vector foeld, again. Commented Jun 22, 2019 at 2:46

I think the question is perhaps confusing, since the term Hopf vector field is usually reserved for your $$X_i$$ vector field (which is tangent to the fibers of the standard Hopf fibration). As I understand, you are instead referring to the vector field which, at a point $$p$$ with $$P(p) = (a,b,c) \in S^2 \subset \mathbb{R}^3$$, is given by $$Y := aX_i + bX_j + cX_k$$. Assuming this is the correct interpretation, then the fiber over $$(1,0,0)$$ has $$Y = X_i$$ and so the fiber of $$P$$ over $$(1,0,0)$$ is a periodic orbit of $$Y$$.