hyperbolic metric on B.

Question

In $\mathbb{R}^{n+1}$ we defined $\langle x,y\rangle = x_1y_1+...+x_ny_n-x_{n+1}y_{n+1}$

The upper sheet of hyperboloid $H$ in $\mathbb{R}^{n+1}$ is the set $\{x \in \mathbb{R}^{n+1} / \langle x, x\rangle = -1 $ and $ x_{n+1} > 0 \}$.

I have the following exercise:

Let be the hyperboloid $H$ in $\mathbb{R}^{n+1}$, consider $P$ the plane $x_{n+1} = 0$ in $\mathbb{R}^{n+1}$ and $B$ the unit ball in $P$

I) If $\phi: H \to P$ is the central projection with center $-e_{n+1}$ prove that $\phi$ is a diffeomorphism, and express $\phi$ in coordinates

I did the computations and found that this map has the following expression $$\phi\left(x_1,\ldots,x_{n+1}\right) = \left(\frac{x_1}{1- x_{n+1}},\ldots,\frac{x_n}{1- x_{n+1}}, 0\right)$$ which shows it is differentiable (because so is any input). The inverse map is $$\phi^{-1}(x_1,\ldots,x_n)= (\lambda x_1,\ldots,\lambda x_n, \lambda-1 )$$ where $\lambda = \frac{2}{1- (x_1^2+\cdots+x_n^2)}$ this shows that $\phi$ is a diffeomorphism.

Now in II) since $\phi$ is an isometry I have to prove that

$\cosh(d(x,y)) = 1 + \frac{2|x-y|^2}{(1-|x|^2)(1-|y|^2)}$ $(1)$

I am stuck at this second part

I know that $\cosh(d(x,y)) = -\langle x,y\rangle$ in $H$. From here, I tried to compute $\phi(1)$ but did not manage to.

Some hint?

Thanks

Answer 1

First, one has a riemannian manifold $(H,g)$ where $g$ is the restriction of $\langle\cdot,\cdot\rangle$ to $TH$. Second, one has a manifold $B$ and a diffeomorphism $$ \phi : H \to B$$ allowing to define a riemannian metric on $B$, say $g_0 = \phi_*g$. By the very definition of the push-forward metric by the diffeomorphism $\phi$, $$ \phi : (H,g) \to (B,g_0) $$ is an isometry. Thus, if $p$ and $q$ are points in $H$, $$ d_g(p,q) = d_{g_0}\left(\phi(p),\phi(q) \right) $$ so one has $$ \cosh\left(d_g(p,q)\right) = \cosh \left(d_{g_0}(\phi(p),\phi(q)) \right) $$ Then, your exercice reduces to the computation, in one hand, of $d_g(p,q)$ in terms of the riemannian metric $g$, and in the other hand, of $d_{g_0}(\phi(p),\phi(q))$ in terms of the euclidean metric on $B$, written $|\cdot|$.

So here are possible hints:

• this can be done computing the differential of $\phi$ (this is possible because you have here an exact expression for $\phi$) and thus the exact value of $g_0 = \phi_*g$ (this can be done thanks to the exact expression of $g$ and $\mathrm{d}\phi$)
• this can maybe be done an easier way: compute the expression $$ 1+2\frac{|\phi(p) - \phi(q)|^2}{\left(1-|\phi(p)|^2\right)\left(1-|\phi(q)|^2\right)} $$ thanks to the exact expression of $\phi$, and where $|\cdot|$ is the usual euclidean norm. You may have an expression in terms of the coordinates of $p$ and $q$ in $\mathbb{R}^{n+1}$ that looks like what you already computed.