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Is there a non vanishing vector field on $S^{3}$ with an infinite family $T_{\lambda}$ of invariant torus such that each $T_{\lambda}$ has an structure of a Kronecker foliation but for every two different $\lambda_{1}$ and $\lambda_{2}$, the corresponding Kronecker foliations are non topological equivalent?

Is there an example of a non vanishing vector field $\tilde{X}$on $S^{3}$ with the above property with the additional condition that $\tilde{X}$ is a lifting of a vector field $X$ on $S^{2}$ with a center singularity via Hopf fibration, see this related question.

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I believe the answer to both of your questions is ''yes".

Any real-analytic vector field on $\mathbb{S}^2$ that has a center singularity can be lifted to a one-parameter family of non-vanishing vector fields on $\mathbb{S}^3$ via the Hopf fibration map so that any vector field from the family has an invariant foliation of tori on which the dynamics is real-analytically conjugate to a linear dynamics on each torus from the foliation and thus, there is an infinite (like the irrational numbers) family of tori with the Kroneker's foliation dynamics of varying angles. (It's kind of like a Liouville-Arnold's theorem).

The three-sphere $\mathbb{S}^3$ is a Lie group and can be seen for example as the unit quaternions (probably the most convenient description) as well as the matrix group $\text{SU}(2)$, which is more or less equivalent to the quaternions. Let $\pi : \mathbb{S}^3 \to \mathbb{S}^2$ be the Hopf fibration map, which is a principle bundle map with an $\mathbb{S}^1$ action on $\mathbb{S}^3$. Let's say $\mathbb{S}^1$ acts on the right, which means the Hopf map can be seen for instance as $\phi \in \mathbb{S}^3 \, \mapsto \, \pi(\phi) = \phi \, i \,\phi^{-1} \in \mathbb{S}^2$ for $i \in \mathbb{S}^2$ in quaternionic notation, which is isomorphic to the classical representation $ \phi = w + j \, z \in \mathbb{S}^3 \, \mapsto \, z/w \, \in \mathbb{C} \cup \{\infty\} \cong \mathbb{S}^2$, also known as $[z:w] \in \mathbb{CP}^1\cong \mathbb{S}^2,$ via stereographic projection. In this picture $\mathbb{S}^1$ is the stabilizer of $i$ and thus its elements are given as the usual $e^{i\theta}$ and their action is simply quaternionic multiplication on the right $\phi \mapsto \phi \, e^{-i\theta}$. The Lie algebra of $\mathbb{S}^3$ is the tangent space $T_1\mathbb{S}^3$ at the identity element $1 \in \mathbb{S}^3$ and is isomorphic to the imaginary quaternions $\text{Im}(\mathbb{H})$. Now use the standard dot product on $T_1\mathbb{S}^3$, which is the same as the Killing form of the algebra, to split it orthogonally into $T_1\mathbb{S}^3 = \mathbb{R}i \oplus E_1$, where $E_1 = \mathbb{R}j \oplus \mathbb{R}k$. Recall that the left translation on the group is $L_{\phi}(\psi) = \phi\psi$ and its tangent map is $(DL_{\phi})_{\psi} : T_{\psi}\mathbb{S}^3 \to T_{\phi\psi}\mathbb{S}^3$ . Then, define the left-invariant vector field $V({\phi}) = (DL_{\phi})_1 i$ which generates the $\mathbb{S}^1$ action on the three-sphere and is thus tangent to the circular fiber of the Hopf bundle (right action is left invariant because right and left commute). Therefore $\ker (D_{\phi}\pi) = \mathbb{R} \, V(\phi)$ is the one-dimensional kernel of the tangent map to the Hopf map. Similarly, define the plane field $E_{\phi} = (DL_{\phi})_1 E_1$ (i.e. a two dimensional distribution or if you prefer a two dimensional subbundle of the tangent bundle of $\mathbb{S}^3$) to obtain the tangent bundle splitting $T_{\phi}\mathbb{S}^3 = \mathbb{R} \, V(\phi) \oplus E_{\phi} = \ker (D_{\phi}\pi) \oplus E_{\phi}$. Hence, the restriction $(D_{\phi}\pi)|_{E_{\phi}} \, : \, E_{\phi} \, \to \, T_{\pi(\phi)}\mathbb{S}^2$ is onto and has no kernel so it is an isomorphism at each point $\phi \in \mathbb{S}^3$. Now, let your real-analytic vector field on the two-sphere $\mathbb{S}^2$ be $X(x)$ and for each $\phi \in \mathbb{S}^3$ define the tangent vector $$Y(\phi) = \Big( (D_{\phi}\pi)|_{E_{\phi}}\Big)^{-1} X(\pi(\phi)).$$ This construction defines a real-analytic lift of $X$, that is $Y(\phi)$ is a real-analytic vector field on the three sphere such that $(D_{\phi}\pi)\, Y(\phi) = X(\pi(\phi))$, which means $Y$ projects down on $X$ via $\pi$. Now, by construction, the distribution $E_{\phi}$ is left-invariant but it is not integrable. It actually defines a contact structure on the three-sphere. The bracket $[V, Y](\phi)$ belongs to $E_{\phi}$ for any $\phi \in \mathbb{S}^3$. Indeed, if $J(\phi)=(DL_{\phi})_1j$ and $K(\phi)=(DL_{\phi})_1k$, then the left-invariant vector fields $J$ and $K$ define a global basis for the distribution $E$. Moreover, $[V, J]=K$ and $[V, K]=-J$. Therefore, $Y(\phi) = \alpha(\phi)J(\phi)+\beta(\phi)K(\phi)$ and thus $$[V, Y] = (\mathcal{L}_V\alpha)J + \alpha[V, J] + (\mathcal{L}_V\beta)K + \beta[V, K] = (\mathcal{L}_V\alpha - \beta)J + (\mathcal{L}_V\beta + \alpha)K.$$ Here $\mathcal{L}_V\alpha$ is the Lie derivative of $\alpha$ in direction of $V$, etc. By construction, $Y$ projects on $X$ and therefore the orbits of $Y$ project onto the orbits of $X$ which are either points, embedded closed circles or immersed real lines. The lifts of the fixed points of $X$ are Hopf circles made of fixed point singularities of $Y$. Let us now focus on the other two types of non-trivial orbits of $X$: embedded circles and immersed real lines (embedded is a special case of immersed). The lift of each of these orbits via $\pi$ on $\mathbb{S}^3$ is either (i) a torus embedded in the three sphere, in the case of a closed orbits, or (ii) immersed cylinder, in the case of an immersed line. Thus we obtain a two-dimensional real analytic foliation on $\mathbb{S}^3$ (with circular singularities), invariant under the $\mathbb{S}^1$ action, i.e. each of its leaves foliates into Hopf fibers over an orbit of $X$. Now, in addition to that, we know that $X$ has center dynamics (possibly local) and thus has an analytic family of closed orbits circling around the center. As already discussed, the lift of each of these orbits via $\pi$ on $\mathbb{S}^3$ is a torus embedded in the three sphere. Thus, we obtain a real analytic foliation (possibly local) of embedded (nested) tori, invariant under the $\mathbb{S}^1$ action, so that each of them is foliated into Hopf fibers over a periodic orbit of $X$ around the center. Denote by $\text{T}_{\phi}$ the unique torus passing thorough a point $\phi \in \mathbb{S}^3$, whenever defined. Also, denote by $\text{S}_{\phi}$ the unique Hopf fiber passing through $\phi \in \mathbb{S}^1$. By construction, both $Y$ and $V$ are tangent to the non-singular leaves of the toric foliation $\text{T}_{\phi}$ (and actually of the total foliation of tori and cylinders obtained by lifting of the orbits of $X$), spanning their tangent planes and forming a plane field (a distribution with singularities) tangent to the foliation. Hence, this distribution is integrable by Frobenius, so $[V, Y](\phi) \, \in \, \mathbb{R} \, Y(\phi) \oplus \mathbb{R} \, V(\phi)$. Consequently, $[V, Y](\phi)$ belongs to the intersection of the transverse (but not complimentary) tangent planes $E_{\phi}$ and $\mathbb{R} \, Y(\phi) \oplus \mathbb{R} \, V(\phi)$. Now, since $\big(\mathbb{R} \, Y(\phi) \oplus \mathbb{R} \, V(\phi) \big) \cap E_{\phi} = \mathbb{R} \, Y(\phi)$, one can immediately conclude that $[V, Y](\phi) = \lambda(\phi)Y(\phi)$ for some real-analytic function $\lambda(\phi)$ defined on $\mathbb{S}^3$ minus the singular Hopf fibers. Finally, apply the map $D\pi$ to the bracket $[V, Y]$ to obtain $$\lambda \, X = \lambda (D\pi) Y = (D\pi)(\lambda Y) = D\pi [V, Y] = [(D\pi)V, (D\pi)Y] = [0, X] = 0.$$ This is possible if and only if $\lambda \equiv 0$. Therefore, $[V, Y] = 0$.

Remark: There could be a more direct argument proving that $[V, Y] = 0$. The Killing form, which is the unique $\text{Ad}$-invariant dot product on the Lie algebra $T_1\mathbb{S}^3$ defines the round metric on $\mathbb{S}^3$ given by left-translations of the Killing form. The $\text{Ad}$-invariance means that the metric is also right-invariant, i.e. it is bi-invariant. As $E_\phi$ is defined as the orthogonal complement of $V(\phi)$, then the splitting, which is defined to be left-invariant, is also right-invariant when acted upon with elements of $\mathbb{S}^1$. Therefore, the lift of $X$ on $E$ via $\pi$ is $\mathbb{S}^1$ invariant, which means that the generating vector field $V$ of the action commutes with $X$.

Recall that the Riemannian metric on the three-sphere is $$(u,v)_{\phi} = ((DL_{\phi^{-1}})_{\phi} \, u, (DL_{\phi^{-1}})_{\phi} \, v)_1$$ for the Killing form/dot product $( \cdot , \cdot )_1$ on the Lie algebra (which is actually the standard dot product in $\mathbb{R}^3$). The metric is bi-invariant (both left and right). The Lie bracket on the Lie algebra $T_1\mathbb{S}^3$ is denoted by $[\cdot, \cdot]_1$ and is actually the standard cross-product in $\mathbb{R}^3$. Although it is the left projection of the vector field Lie bracket of left-invariant vector fields, it is not equal to the projection of the vector field Lie bracket of non-invariant vector fields. That's why the superscript $1$, in order to distinguish it from the vector field bracket.

Define the vector field $$\tilde{W}(\phi) = (DL_{\phi})_1 \Big(\text{ad}(i) \circ (DL_{\phi^{-1}})_{\phi} \, Y(\phi) \Big) = (DL_{\phi})_1 \, [i, (DL_{\phi^{-1}})_{\phi} \, Y(\phi)]_1. $$ By construction of $Y$, the new field $\tilde{W}$ is nonzero, whenever $Y$ is non-zero and is orthogonal to both $Y$ and $V$. Next, normalize it to the unit vector field $W(\phi) = \tilde{W}(\phi) / \|\tilde{W}(\phi)\|$ defined on the three-sphere minus some Hopf fiber singularities. It's an analytic unit vector field orthogonally transverse to the toric foliation $\text{T}_{\phi}$. We assume it is pointing inwards towards the singularity of the toric foliation (if not, just take $-W$). Let $\phi_0(s) \, : \, s \in (L,\infty)$ be one integral curve of $W$, i.e. $\dot{\phi}_0(s) = W(\phi_0(s))$, with the property that its projection $\pi(\phi_0(s))$ on $\mathbb{S}^2$ converges towards the central singularity of $X$ when $s \to \infty$. By construction, $\phi_0(s)$ is transverse to each $\text{T}_{\phi}$ and intersects it at exactly one point. Hence, we can rewrite the analytic foliation $\text{T}_{\phi}$ as a real-analytic one parameter family of tori $\text{T}_{s} = \text{T}_{\phi_0(s)}$.

Denote by $\Psi^{t_1}(\phi)$ the phase flow of the vector field $Y$, i.e. $$\frac{d}{d t_1} \Psi^{t_1}(\phi) = Y\big(\Psi^{t_1}(\phi)\big)\, , \,\,\, \Psi^0(\phi) = \phi.$$ Observe that $\phi e^{i t_2}$ is the phase flow of $V$ by construction. Furthermore, define the real analytic map $F \, : \, \mathbb{R}^2 \times (L,\infty) \, \to \, \mathbb{S}^3$ by $$F \, : \, (t,s) = (t_1,t_2,s) \, \mapsto \, \Psi^{t_1}\big(\phi_0(s)\, e^{it_2}\big) = \Psi^{t_1}\big(\phi_0(s)\big) \, e^{it_2},$$ where the last equality holds due to the commutation between $X$ and $V$. A straightforward computation shows that \begin{align*} (D_{t,s}F)\frac{\partial}{\partial t_1} &= Y(F(t,s))\, , \\ (D_{t,s}F)\frac{\partial}{\partial t_2} &= V(F(t,s))\, , \\ (D_{t,s}F)\frac{\partial}{\partial s} &= (DR_{e^{-it_2}}) \circ (D\Psi^{t_1})\, W(\phi_0(s)) = (D\Psi^{t_1}) \circ (DR_{e^{-it_2}}) \, W(\phi_0(s)), \end{align*} all three of them being linearly independent. That mean that $F$ has a maximal rank and is therefore a real-analytic local diffeomorphism. It is onto its image and as such is actually a universal covering map onto $F\big(\mathbb{R}^2 \times (L, \infty)\big)$. It also shows that $F$ maps the straight constant vector field $\partial/\partial t_1$ to $Y$ and $\partial/\partial t_2$ to $Y$.

One can see that $s$ can be expressed as a function of $\phi$ as follows. For $\phi \in \mathbb{S}^3$, take the unique torus $\text{T}_{\phi}$ from the toric foliation (as long as $\phi$ is in the domain of the toric foliation). By construction, it intersects the curve $\phi_0(s)$ at a unique point, determined by a unique $s$. Thus, set $s=s(\phi)$. The function $s(\phi)$ is real analytic because it is the $s-$coordinate function of the inverse map $F^{-1}$ (existing locally). It is also a first integral of $Y$ determining the toric foliation as its level surfaces.

Let $\delta_s(t_1)$ be an integral curve of $Y$ passing through the point $\phi_0(s)$, i.e. $\dot{\delta}_s(t_1) = Y\big(\delta_s(t_1)\big)$ and $\delta_s(0) = \phi_0(s)$. It lies on $\text{T}_s$. Denote by $\delta_s$ the part of the orbit $\delta_s(t_1)$ which starts from $\phi_0(s)$ and goes around the torus until it comes back and intersect the Hopf fiber $\text{S}_{\phi_0(s)}$ again for the first time (not counting the initial staring point $\phi_0(s))$. In other words, this is a Poincar\'e map construction. Denote the first return point by $\phi_1(s)$. Now let $\gamma_s$ be the simple arc on the Hopf fiber $\text{S}_{\phi_0(s)}$ starting from $\phi_1(s)$ and ending at $\phi_0(s)$ following the orientation provided by the vector field $V$ which is tangent to the fiber. Consequently, we have a (piecewise smooth) simple closed loop $\delta_s \cup \gamma_s$ on the torus $\text{T}_s$ from $\phi_0(s)$ to itself. Since $F$ is a universal covering map, by the path lifting property of covering spaces, lift $\delta_s \cup \gamma_s$ on the universal cover $\mathbb{R} \times (L,\infty)$, which actually ends up being a curve on $\mathbb{R} \times \{s\}$, starting from $(0,0,s)$ to the end point $(a_1(s),a_2(s), s) \neq (0,0,s)$ depending real-analytically on $s$. Define the tangent vector $a(s)=(a_1(s),a_2(s),0)$ at $(0,0,s)$. If we denote by $e_2$ the vector $2\pi \partial/\partial t_2$ or in coordinates $(0,2\pi,0)$ at $(0,0,s)$, then $F \, : \, \left(\mathbb{R}^2 / \big(\mathbb{Z} \, a(s) \oplus \mathbb{Z} \, e_2 \big) \right) \times \{s\} \, \to \, \text{T}_s$ is a real analytic diffeomorphism between tori. Then, let us change the coordinates by the linear transformation \begin{align*} t_1 &= a_1(s) \sigma_1 \\ t_2 &= a_2(s) \sigma_1 + 2\pi \sigma_2. \end{align*} Then one goes from $(t,s) = (t_1,t_2,s)$ coordinates to $(\sigma, s) = (\sigma_1, \sigma_2, s)$ coordinates. Denote by $A(s)$ the inverse linear map of that transformation, i.e. $A(s)$ is given by \begin{align*} \sigma_1 &= \frac{1}{a_1(s)} t_1 \\ \sigma_2 &= \frac{a_2(s)}{2\pi a_1(s)} t_1 + \frac{1}{2\pi} t_2. \end{align*} For simplicity, let $\omega_1(s) = 1/a_1(s)$ and $\omega_2(s) = a_2(s)/(2\pi a_1(s))$. Then it is immediate to see that under the transformation $A(s)$ we can rewrite the vector fields as follows: $$\frac{\partial}{\partial t_1} = \omega_1(s) \frac{\partial}{\partial \sigma_1} + \omega_2(s) \frac{\partial}{\partial \sigma_2}\, \, \,\, \text{ and } \,\,\,\, \frac{\partial}{\partial t_2} = \frac{1}{2\pi} \frac{\partial}{\partial \sigma_2}.$$ As $a_1(s)$ and $a_2(s)$ depend real analytically on $s$, so do $\omega_1(s)$ and $\omega_2(s)$ (the denominator $a_1(s)$ is always positive so never zero, because it is actually the time of first return of the Poincar\'e map). Now, define the new real-analytic map (a local real-analytic diffeomorphism and a universal covering map onto its image) \begin{align*} \Phi \, &: \, \mathbb{R}^2 \times (L,\infty) \, \to \, \mathbb{S}^3 \\ \Phi \, &: \, (\sigma, s) \, \to \, F\big( A(s)\sigma, s \big). \end{align*}

Finally, let us define the one parameter family of vector fields $\tilde{X}_{\varepsilon}(\phi) = Y(\phi) + \varepsilon \, V(\phi)$, then by construction, $(D_{\phi}\pi)\big(Y(\phi) + \varepsilon \, V(\phi)\big) = X(\pi(\phi))$ for any $\phi \in \mathbb{S}^2$. The new vector field $\tilde{X}_{\varepsilon}$ is non-vanishing on the three-sphere for $\varepsilon \neq 0$ and is tangent to the toric foliation $\text{T}_{\phi}$. Also, written in $(\sigma, s)$ coordinates, it becomes $$(D_{\phi}\Phi^{-1}) (\tilde{X}_{\varepsilon}(\phi)) = (D_{\phi}\Phi^{-1}) \big(Y(\phi) + \varepsilon \, V(\phi)\big) = \omega_1(s) \frac{\partial}{\partial \sigma_1} + \left(\omega_2(s) + \frac{\varepsilon}{2\pi}\right) \frac{\partial}{\partial \sigma_2}.$$ If we set $\omega_2(s,\varepsilon) = \omega_2(s) + \varepsilon/{2\pi}$, its orbits satisfy the equations \begin{align*} \dot{\sigma}_1 &= \omega_1(s) \\ \dot{\sigma}_2 &= \omega_2(s,\varepsilon)\\ \dot{s} &= 0. \end{align*} which means that its solutions are $\big(\sigma_1 + \omega_1(s) t, \,\, \sigma_2 + \omega_2(s, \varepsilon) t, \,\, s \big)$ on the space $\Big(\mathbb{R}^2 / 2\pi\mathbb{Z}^2\Big) \times (L,\infty)$. Since by construction and due to the $\varepsilon$-freedom $\omega_2(s,\varepsilon)/\omega_1(s)$ is not constant, we can conclude that on infinitely many foliating tori $\text{T}_s$ the orbits of $\tilde{X}_{\varepsilon}$ exhibit the dynamics of a Kroneker foliation so that generically, the Kroneker dynamics on two such tori is non-equivalent.

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  • $\begingroup$ Phew, very interesting answer. It's going to take me some time to understand it. $\endgroup$
    – John Small
    Apr 20, 2021 at 19:06

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