Give an example of a flow on the projective plane RP^ 2 such that there is exactly one fixed point and all other orbits are periodic.

Question

The rough idea I had was that RP^2 is the collection of all lines in R^3 passing through the origin, so if we rotate each line by some angle, then the origin is a fixed point and all other points have periodic orbits.

Don't know how to formally proceed though.

Thanks for the help!

Answer 1

You can use the fact that the projective plane is equivalent to a sphere with antipodal points identified. Thus it is sufficient to find a flow on $S^2$ which is invariant under reflection (so that it lifts to projective space unambigously) and has two (antipodal) fixed points and all other orbits periodic. Can you do it? Hint: there is a particularly simple example with nice geometric interpretation.