It is not the first time that I come across the following metric for the unit sphere $S^d$: $$g_{mn} = \delta_{mn} + \frac{y^my^n}{1-y^py^p},$$ where $m,n,p=1,\dotsc,d$ and we are summing over $p$.

What is unclear to me is which coordinates are these $y^m$. They are neither spherical nor stereographic, for evident reasons. However, I noticed that the components of the metric are remarkably similar to the Fubini-Study metric on the complex projective space in homogeneous coordinates.

Among the places where I found these expression are: the famous review "Kaluza-Klein Supergravity" by Duff, Nilsson and Pope, and the recent article https://arxiv.org/abs/arxiv:1410.8145. The question then is: how are these coordinates defined?

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    $\begingroup$ I think this is just the euclidean metric pulled back to the surface defined by $y^m y_m = 1$. I don't know the nomenclature of these coordinates but you just eliminate one of the $y$'s. $\endgroup$
    – emir sezik
    Jul 18, 2023 at 16:55
  • $\begingroup$ Right! That's like projecting "straight" onto a d-1 dimensional plane. I had implicitly excluded those because of the thought that they were conformal, but evidently that's not the case $\endgroup$
    – User175a23
    Jul 22, 2023 at 18:54

1 Answer 1


As emir sezik said in a comment, this is the metric you get if you start with the Euclidean metric in $d+1$ dimensions with coordinates $(y^1, \ldots, y^d, z)$ and substitute $z = \sqrt{1 - \mathbf y^2}$. The coordinates cover only one hemisphere.


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