A symplectic look at surfaces of revolution. (Q1419563)

Given a surface of revolution in 3-space, there exist isothermal \(S^1\)-equivariant coordinates \((t,\theta)\) such that the metric is \(e^{\psi(t)}(dt^2+d\theta^2)\). Replacing \(t\) by the moment map \(\tau:=\int e^{\psi(t)}\,dt\) we get action angle coordinates \((\tau,\theta)\), and a function \(\phi(\tau)=e^{\psi(t)}\) called the momentum profile, that describes the length \(2\pi\sqrt{\phi(\tau)}\) of the orbit \(\tau= \text{const}\). The Gaussian curvature in these coordinates is simply given by \(K=-\frac 12\phi''(\tau)\). The author describes this construction with gentle care. He shows conversely that any reasonable positive \(C^2\)-function \(\phi\) on an interval leads to a momentum map \(\tau\) on an anulus of the Riemann sphere and to an abstract surface of revolution (that may not admit a realization in 3-space). He discusses singularities, completeness and Calabi metrics of these.