Geodesics of Poincaré Ball Model

Question

I am asked to consider the Poincaré ball model of 3d hyperbolic space and characterize its geodesics. I.e. we have a pseudo-riemannan manifold $(M,g)$ where $$ M = \{(x^1,x^2,x^3)\in \mathbb{R}^3 : \|x\|_{eucl}<1\} $$ is equipped with the metric \begin{align*} g=4\left(\frac{1}{1-\|x\|^2}\right)^2 (dx^1 \otimes dx^1+dx^2\otimes dx^2 + dx^3\otimes dx^3). \end{align*}

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More precisely,

1. I am asked to show that the lines through the origin in $\mathbb{R}^3$ are geodesics (by considering appropriate isometries) - I have no clue how to start here.

2. Using this I should somehow figure out how to show that the geodesic $c_V$ with $c_V(0)=0$ $\dot c_\xi (0) = V$ is given by \begin{align*} c_V(t) = \tanh(|V|t) \frac{V}{|V|}. \end{align*}

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I do not know how to show 1. Neither do I have any clue of how to go from 1. to 2. I would appreciate any help I can get with this.

Answer 1

The fixed point set of an isometry is totally geodesic. See if you can show that rotations about lines passing through the origin are isometries.

Then solve the geodesic equation in one dimension along that line to arrive at a geodesic parametrization. (You can wlog assume that $V = (1, 0, 0)$ --- do you see why?)

Answer 2

Probably the $\tanh$ has something to do with relation between the euclidean radius and the hyperbolic radius. If the hyperbolic radius is $R$, then the Euclidean radius is $\tanh(R/2)$.