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Came across an example of Riemannian metric example today, one of them was of a Riemannian metric defined on upper half plane $ \mathbb{H^{2}} = \{ (x,y): y>0\} $ as:

$$ ds^2 = \frac{dx^2 + dy^2} {y^2}$$

I have been thinking how to go about finding this, but no success! My idea was to first find $g_{ij}$, the inner product of tangent vectors like I did in case of finding the same type of expression for Euclidean Riemannian metric but there I knew what tangent vectors for $R^n$ looked like but no idea here!

Can someone help? Thanks.

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  • $\begingroup$ So is your question how to understand this notation. Or what to you mean by "figure out"? $\endgroup$ – MaoWao Jan 25 at 13:26
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    $\begingroup$ Indeed, it should be $g_{xx}=g_{yy}=1/y^2$ and $g_{xy}=0$. $\endgroup$ – Jon Jan 25 at 13:29
  • $\begingroup$ @Jon: I guessed as much but how did we get $g_{xx} , ..$ sorry if that's a dumb thing to ask and something that I am expected to know! $\endgroup$ – Mojojojo Jan 25 at 13:47
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    $\begingroup$ $ds^2 = g_{xx} \ dx^2 + 2 g_{xy} \ dx dy + g_{yy} \ dy^2$. $\endgroup$ – Spencer Jan 25 at 13:53
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    $\begingroup$ @clear, if you our question is "how do I get the components of the metric tensor from the given expression?" Then you just read off the appropriate coefficients. As it stands your question is a bit unclear. $\endgroup$ – Spencer Jan 25 at 13:54

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