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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?

Arctic Char
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Shreya
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  • So is your question how to understand this notation. Or what to you mean by "figure out"? – MaoWao Jan 25 '19 at 13:26
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    Indeed, it should be $g_{xx}=g_{yy}=1/y^2$ and $g_{xy}=0$. – Jon Jan 25 '19 at 13:29
  • @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! – Shreya Jan 25 '19 at 13:47
  • I think we need to first know, given a coordinate chart $(U, (x,y))$ around a point $p$, how to find the vectors in $T_{p}\mathbb{H^{2}}$...now what? – Shreya Jan 25 '19 at 13:52
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    $ds^2 = g_{xx} \ dx^2 + 2 g_{xy} \ dx dy + g_{yy} \ dy^2$. – Spencer Jan 25 '19 at 13:53
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    @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. – Spencer Jan 25 '19 at 13:54
  • It is possible that what the OP means is "How do I use this expression to evaluate the metric on a pair of tangent vectors at a point", but yeah, the OP needs to clarify the question. – Lee Mosher Jan 25 '19 at 14:00
  • If that's what is meant, however, then I would say that this is a duplicate of https://math.stackexchange.com/questions/294792/what-does-ds2-mean-and-how-does-it-specify-a-metric?rq=1 – Lee Mosher Jan 25 '19 at 14:01
  • I meant to ask, how do I find that expression for $ds^2$? – Shreya Jan 26 '19 at 08:39
  • Possibly relevant: Pseudosphere – ccorn Oct 10 '20 at 20:35

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