Geodesic Flow on Global Holomorphic Sections of ${TS}^2$

Abstract

We study the geodesic flow on the global holomorphic sections of the bundle $\pi:{\mbox{TS}}^2\rightarrow \mbox{S}^2$ induced by the neutral Kähler metric on the space of oriented lines of ${\Bbb{R}}^3$, which we identify with ${\mbox{TS}}^2$. This flow is shown to be completely integrable when the sections are symplectic, and the behaviour of the geodesics is described.