I want to construct free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$.

For $n=2m-1$, consider $S^n ⊂ C^m$. Then $S^1$ freely act on $S^n$ by $(ξ, (z_1 , z _2 , · · · , z _m )) → (ξz_1 , ξz_2 , · · · , ξz_m )$ where $ξ ∈ S^1 $. Then $S^1/\mathbb{Z}_2\cong S^1$ will act freely on $\mathbb{R}P^n = S^n /\mathbb{Z}_2$.

I want to know what are all other free $S^1$ action on $\mathbb{R}P^n$?

Also How to construct a free $S^1$ action on $\mathbb{C}P^n$? Thank you in advance...

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