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Let $\mathbb{S}^{2n+1}$ be the Euclidean round sphere of radius 1 and let $\mathbb{C}P^n$ be the complex projective space endowed with the Fubini-Study metric, obtained as the quotient space of that sphere by an $\mathbb{S}^1$ action. Let $\pi : \mathbb{S}^{2n+1} \to \mathbb{C}P^n$ be the projection ($\pi$ is a Riemannian submersion).

I read in another post (see the answer of this post) that any geodesic in $\mathbb{C}P^n$ lifts to a (unique, when a point on the fibre is chosen) geodesic in $\mathbb{S}^{2n+1}$ which is everywhere orthogonal to the fibres. That said, how can we relate the distance of two points $\tilde{p}$ and $\tilde{q}$ on the sphere with the distance of $\pi(\tilde{p})$ and $\pi(\tilde{q})$ on $\mathbb{C}P^n$?

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    $\begingroup$ If it's no trouble, could you please provide a link to the claim about geodesics lifting to geodesics orthogonal to the fibres? (Assuming that claim is correct, there's still the issue of how to scale the Fubini-Study metric; do you want the projection to be a Riemannian submersion?) $\endgroup$ – Andrew D. Hwang Jun 29 '17 at 2:55
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    $\begingroup$ Provided. I also added the Riemannian submersion hypothesis. $\endgroup$ – Eduardo Longa Jun 29 '17 at 3:28
  • $\begingroup$ You can't assume that the map is Riemannian, it either is Riemannian or not. $\endgroup$ – Mariano Suárez-Álvarez Jun 29 '17 at 3:43
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    $\begingroup$ The metric on the complex projective space is such that the projection is a Riemannian submersion. $\endgroup$ – Eduardo Longa Jun 29 '17 at 3:48
  • $\begingroup$ Modulo details, if $\ell(\tilde{p})$ and $\ell(\tilde{q})$ denote the complex lines in $\mathbf{C}^{n+1}$ spanned by $\tilde{p}$ and $\tilde{q}$ respectively, the distance between their images $\pi(\tilde{p})$ and $\pi(\tilde{q})$ is the infimum of the angles between $\ell(\tilde{p})$ and $\ell(\tilde{q})$. If memory serves, this is in Kobayashi-Nomizu. Offhand I don't have a clean formula, or a detailed proof of the claim about geodesic lifting, hence this comment rather than an answer. $\endgroup$ – Andrew D. Hwang Jun 29 '17 at 10:02

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