Dirac flow on the 3-sphere

Abstract

We illustrate some well-known facts about the evolution of the 3-sphere (S³, g) generated by the Ricci flow. We define the Dirac flow and study the properties of the metric \(\bar g = dt^2 + g(t)\), where g(t) is a solution of the Dirac flow. In the case of a metric g conformally equivalent to the round metric on S³ the metric \(\bar g\) is of constant curvature. We study the properties of solutions in the case when g depends on two functional parameters. The flow on differential 1-forms whose solution generates the Eguchi–Hanson metric was written down. In particular cases we study the singularities developed by these flows.