6
$\begingroup$

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?

$\endgroup$
2
  • 1
    $\begingroup$ 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

8
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.