How to show that a map is an isometry I'm having a difficulty understanding how to go on proving a certain map is an isometry. It should be really basic and simple, but for some reason I can't understand how to do this..

The situation is this: I have 2 manifolds, $D,M$: 


*

*$D$ is the Poincare disk $\{x\in\mathbb{R}^{n+1}:x_0=0,\sum_{i=1}^n{x_i}<1\}$, and 

*$M$ is the positive-half-space, $\{x\in\mathbb{R}^n:x_n>0\}$).


Each has its own metric:


*

*for $D$: $g_{ij}^{(1)} = \frac{4\delta_{ij}}{1-\sum_\alpha u_\alpha^2}$.

*for $M$: $g_{ij}^{(2)} = \frac{\delta_{ij}}{x_n^2}$.


And I have a map, $f:D\to M$, given by $f(p) = 2\frac{p-p_0}{|p-p_0|^2}+p_0$ where $p_0=(0,\ldots,0,-1)\in\mathbb{R}^n$.

By my understanding of the definition of isometry, I should take 2 vectors, $p_1,p_2\in D$, and show that $g^{(1)}(p_1,p_2)=g^{(2)}(f(p_1),f(p_2))$.
From here I'm really confused.. I have $g_{ij}$ (not $g$), which is defined for $T_pD$, and we get $g$ from $g_{ij}$ (though how exactly I'm not sure). What I thought is that I need to find $df$, and show that for a basis $\partial_i$ of $T_pD$, $g_{ij}^{(1)}(\partial_i,\partial_j)=g_{ij}^{(2)}(df_p(\partial_i),df_p(\partial_j))$, but I'm really not sure on how to do this, and I'll be glad if you could give me a direction..
Thanks
 A: Using your notation, we can get $g$ from $g_{ij}$ by integrating along paths. Given $\gamma:[0,1] \to M$ with $\gamma(p_1)=0, \gamma(p_2)=1$, its arclength can be computed as
$$L(\gamma)=\int_0^1 g_{ij}(\gamma'(t),\gamma'(t)) \, dt \, .$$
Then $g(p_1,p_2)$ is defined to be $\inf_{\gamma(0)=p_1,\gamma(1)=p_2} L(\gamma)$.
So your last sentence is correct; if you can show that $df$ preserves the inner product on $TD$, that will imply that $f$ preserves arclengths of curves and thus also the metric-space structure on $D$. In fact, "$df$ preserves the inner product" is generally taken as the definition of an isometry between Riemannian manifolds, since the actual metric-space structure can be difficult to work with directly (being an infimum over an infinite-dimensional space of curves).
So you just need to be able to compute $df_p$ for all $p$; to do this, remember that, once you've chosen local coordinates for your source and target manifolds, $df$ becomes the Jacobian of $f$. The problem statement hands you a set of coordinates (and in fact gives you your two metrics in those coordinates), so...
