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I'm trying to give a few examples of symmetric manifolds. In the article "Introduction to Symmetric Spaces and Their Compactification" Lizhen Ji mentions the Poincaré disk as a symmetric space in the following way:

[itex]D = \{z \in \mathbb C | |z| < 1\}[/itex]

with metric

[itex]ds^2 = \frac{|dz|^2}{(1-|z|^2)^2}[/itex].

First question: I don't know this "ds²" notation and I wasn't able to figure out how to convert it into the more familiar notation [itex]g_p(x, y) = ...[/itex]. Is this possible? Last semester we defined a Riemannian metric on the Poincaré disk as follows:

[itex]g_p(x, y) = \frac{4\langle x, y\rangle}{(1-||p||^2)^2}[/itex]

Is this the same metric?

The second thing:

Ji says that the group

[itex]SU(1, 1) = \{\begin{pmatrix}a && b \\ \bar b && \bar a\end{pmatrix} | a, b \in \mathbb C, |a|^2-|b|^2 = 1\}[/itex]

acts isometrically and transitively on D by setting

[itex]\begin{pmatrix}a && b \\ \bar b && \bar a\end{pmatrix}z = \frac{az+b}{\bar b z+\bar a}[/itex]

But he doesn't prove this and instead says "This follows by a direct computation".

I would like to do this computation, and as I see this I need to show:

[itex]g_{Az}(A_*x, A_*y) = g_z(x, y)[/itex]

But then again I would need to know the metric explicitly and also unfortunately I couldn't figure out how to compute the derivative [itex]A_*x[/itex] for this group action :(

I would be very grateful for some help with this. Thank you in advance!

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# Poincaré disk: metric and isometric action

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