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