== (A) S^3
Xr, Yr tangent to S^3 (orthogonal to grad rho): True True
Yr = J0 Xr: True
sigma_* Xr = Xr o sigma: True
sigma_* Yr = -Yr o sigma (anti-CR: sigma_* J = -J sigma_*): True
|Xr|^2 = |Yr|^2 = rho, <Xr,Yr> = 0 (Euclidean metric on H is Hermitian, (Xr,Yr) orthogonal frame): True True True
theta(Xr) = theta(Yr) = 0: True True
sigma^* theta = -theta: True
Levi form d theta(Xr, J0 Xr) = 2 rho > 0: True
(z,w)->(conj z, w): pullback of theta is [y1, -x1, -y2, x2] ; minor pull^theta (x1,y1 part) = 0 (nonzero => not contact)
== (B) Heisenberg (x,y,t), theta = dt + 2(x dy - y dx)
theta(X)=theta(Y)=0: True True
tau_* X = X o tau: True
tau_* Y = -Y o tau: True
tau^* theta = -theta: True
[X,Y] = [0, 0, -4] ; d theta(X,Y) = -theta([X,Y]) = 4
tau maps horizontal piecewise-flow paths to horizontal paths of the same length, endpoint tau(q) (200 trials): True
== (A') the non-example (z,w)->(conj z, w): does it map H_p into H_{sigma2 p}?
theta_{sigma2 p}(d sigma2 Xr(p)) = 2*x1*y2 + 2*x2*y1  (not identically 0 on S^3 => not a contact map, so not conformal)
