Cut locus of $\mathbb{CP}^n$

Question

I can show that the cut locust of some $p\in\mathbb{RP}^n$ is just a copy of $\mathbb{RP}^{n-1}$ coming from an equatorial $S^{n-1}$ sphere under the projection $S^n\mapsto\mathbb{RP}^n$. I know that for $p\in\mathbb{CP}^n$ you are supposed to get $\mathbb{CP}^{n-1}$ but a just don't quite see it. How would one prove this?

Answer 1

For tongue in cheek answer, the farthest points away from $[1:0:0:\ldots :0]$ are clearly those of the form $[0:z_1:\dots: z_n]$, i.e., a $\mathbb{C}P^{n-1}$.

To make this rigorous, let $\pi:S^{2n+1}\rightarrow \mathbb{C}P^n$ be the projection. Then, I'd prove it by the following claims.

Claim 1: For every geodesic $\gamma$ on $\mathbb{C}P^{n-1}$ starting at $[1:0:\ldots:0]$, there is a unique geodesic $\tilde{\gamma}$ on $S^{2n+1}$ starting at $(1,0,\ldots 0)$ with tangent vector orthogonal to $(i,0,\ldots ,0)$ with $\pi(\tilde{\gamma})=\gamma$. (In fact, $\tilde{\gamma}$ is nothing but the horizontal lift of $\gamma$.)

Claim 2: The geodesic $\gamma$ stops minimizing when $\tilde{\gamma}$ is a quarter of the way around $S^{2n+1}$.

Claim 3: The collection of all "stopping points" of such $\tilde{\gamma}s$ is $S^{2n-1} = \{(0, z_1,...,z_n)\in S^{2n+1}\}.$ The Hopf circle action on $S^{2n+1}$ restricts to the Hopf action on this $S^{2n-1}$.

Claim 4: The cut locus of $\mathbb{C}P^n$ relative to $[1:0:\ldots:0]$ is given by $\pi(S^{2n-1})$,that is $S^{2n-1}/S^1 = \mathbb{C}P^{n-1} =\{[0:z_1:\ldots:z_n]\in\mathbb{C}P^n\}.$