Dirac flow on the $3$-sphere

We illustrate some well-known facts about the evolution of the $3$-sphere $(S^3,g)$ generated by the Ricci flow. We define the Dirac flow and study the properties of the metric $\overline g=dt^2+g(t)$, where $g(t)$ is a solution of the Dirac flow. In the case of a metric $g$ conformally equivalent to the round metric on $S^3$ the metric $\overline g$ 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.