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I am trying to construct an n-parameter family of measure preserving real analytic diffeomorphisms on $S^3$ which preserves the $S^1$ fibres of the Hopf fibration but acts transitively on the image $S^2$ more precisely:

The Hopf fibration $p:S^3\to S^2$ is defined by $$(z_1,z_2)\mapsto (2z_1\overline{z_2},|z_1|^2-|z_2|^2)$$ I am trying to construct an n parameter family of measure preserving real analytic diffeomorphisms $\{f_{(\theta_1,\ldots,\theta_n)}\}_{(\theta_1,\ldots\theta_n)\in T^n}$ $$f_{(\theta_1,\ldots,\theta_n)}:S^3\to S^3$$ Satisfying the following two properties:

(1) Given any $x,y\in S^3$, there exists $(\theta_1,\ldots,\theta_n)\in T^n$ such that

$$p(f_{(\theta_1,\ldots,\theta_n)}(x))=p(y)$$

(2) given any $x\in S^3$ and $\phi_t((z_1,z_2)):=(e^{2\pi i t}z_1,e^{2\pi i t}z_2)$ then $\forall\;(\theta_1,\ldots,\theta_n)\in T^n$,

$$f_{(\theta_1,\ldots,\theta_n)}\circ\phi_t(x)=\phi_t\circ f_{(\theta_1,\ldots,\theta_n)}(x)$$

I know how to do it if the second condition is removed. Not sure if it could be done with both conditions intact. $n$ can be any natural number. Smaller the better.

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  • $\begingroup$ Your second condition doesn't make much sense, because $f_{\theta_i}$ is a diffeomorphim, so $x = y$. Please reformulate the question. $\endgroup$
    – Daniele Zuddas
    Commented Feb 2, 2014 at 6:55
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    $\begingroup$ Maybe you want something else, but, why not take the $3$-parameter group of isometries of $S^3$ that consists of left multiplication by elements of $\mathrm{SU}(2)$ (these commute with $\phi_t$). That has your desired properties. If you want your family to be parametrized by the $3$-torus, just take any real analytic mapping $f:T^3\to SU(2)$ that is surjective, and you are done. It can be done with $n=2$ if you want to work a little harder, but you can't get down to $n=1$, obviously. $\endgroup$
    – Robert Bryant
    Commented Feb 2, 2014 at 13:25

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