# Mysterious Coordinates for Metric on a Sphere

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?

• 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. Jul 18, 2023 at 16:55
• 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 Jul 22, 2023 at 18:54

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.