Can anyone provide an explicit contactomorphism between the following two contact structures on the circle cross the plane?

1) The standard contact structure on threespace, but with the line that fibers over the origin of the front plane removed.

2) The cosphere bundle of the plane.


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    $\begingroup$ I'm not sure that I'm familiar with your terminology. Is your example 1, simply $\mathbb{R}^3$ with coordinates $(x,y,z)$, and contact form $\mathrm{d}y-z\,\mathrm{d}x$ after one removes the line $x=y=0$? $\endgroup$ – Robert Bryant Mar 24 '15 at 21:08
  • $\begingroup$ Yes, precisely, Robert. The projection to the xy plane is called the "front projection." More generally, we can project any first jet bundle $J^1(X)$ to $X\times \mathbb R.$ (I tried to write a coordinate-free question, but at the expense of using terminology, I guess. The Jargon Alternative?) $\endgroup$ – Eric Zaslow Mar 24 '15 at 23:53

This answer comes from Sheng-Fu Chiu at Northwestern.

Let $$\alpha = dz - ydx$$ be the standard contact form on $\mathbb R^3$. Let $a = z/x - y$. Let $\tan(\theta)$ be the unique real solution of $T^3 - aT^2 + 3T - a = 0,$ with branch chosen so that $\cos(\theta)$ and $x$ have the same sign.

Put $r = x/\cos(\theta), t = y + 3\sin(\theta)\cos(\theta),$ and set $R = \log(r).$ These definitions invert the coordinate transformation $$(x,y,z) = (rc,t-3sc,rs^3 + trc),$$ where $c = \cos(\theta)$ and $s = \sin(\theta)$ and $r>0$.

Put $\rho = \sqrt{(3s-2s^3)^2 + c^2}$ and define $\phi$ by $\cos(\phi) = (3s-2s^3)/\rho, \sin(\phi) = c/\rho.$ Then $$\alpha = \frac{e^R}{\rho}(\cos(\phi)dR + \sin(\phi)dt),$$ which defines the standard contact structure on the cosphere bundle of the $(R,t)$ plane.


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