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The equation for a 2-dimensional hyperboloid is (look here):

$$R^2=x^2+y^2-z^2$$

The metric is given by:

$$\mathrm{d}s^2=R^2((\cosh^2\theta+\sinh^2\theta)\mathrm{d}\theta^2+\cosh^2\theta \mathrm{d}\phi^2)$$

I'm interested in the metric of a 4-dimensional hyperboloid ($R^2=v^2+w^2+x^2+y^2-z^2$). Is there some general procedure to extend the metric to a higher dimension?

Math Attack
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Il Guercio
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    You're going to need to start with coordinates and a metric on the unit $3$-sphere. – Ted Shifrin Sep 19 '23 at 15:28
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    Asking about "the" metric on a 4-dimensional hyperboloid, or even on the 2-dimensional one, does not make any sense. There are *a lot* of metrics on hyperboloids. You have not even mentioned the obvious one $dx^2 + dy^2 + dz^2$ on the 2-dimensional hyperboloid. Can you be more specific regarding what kind of metric you need? – Lee Mosher Sep 19 '23 at 15:35
  • @LeeMosher OP has given the pullback of the Euclidean metric to the hyperboloid in (global) local coordinates on the hyperboloid. We assume (perhaps wrongly) that OP is looking for the induced metric on the hyperboloid in $\Bbb R^5$, in terms of coordinates on the hyperboloid. – Ted Shifrin Sep 19 '23 at 16:16
  • Perhaps, but, I'd like to hear that from the OP. – Lee Mosher Sep 19 '23 at 19:03
  • @LeeMosher What I'm interested in is the extension of the stuff mentioned in the article to four dimensions. Or maybe first to three dimensions. I have seen such an extension for the 2-sphere to higher dimensions, so I guess it's possible for the 2D hyperboloid too. – Il Guercio Sep 22 '23 at 06:18

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