Every point on the hyperboliod is mapped via the stereographic projection to the Poincaré ball.
Now, how does one map tangent vectors on the hyperboloid to tangent vectors on the Poincaré ball?
One approach that I thought about was to use the same stereographic projection. However, what I‘m not sure about is whether the stereographically projected tangent vectors on the hyperplane, where the Poincaré ball lies, are of the correct scale. Certainly, they will point in the right direction.
What is a principled approach to map tangent vectors from one manifold to another manifold if there exists an isometry between them (which in this case is the stereographic projection)?
Stereographic Projection
Let $R$ be the radius of the $n$-dimensional hyperboloid, and $K$ be the sectional curvature. Then we have $R=\frac{1}{\sqrt{|K|}}$. The the formula for the stereographic projection is:
$$ \begin{align*} \text{sproj}\colon&&\mathbb{R}^{n+1}&\to\mathbb{R}^{n} \\ && (x_1,\ldots,x_n,x_{n+1}) &\mapsto \frac{1}{1-\sqrt{|K|}x_{n+1}} (x_1,\ldots,x_n) \end{align*} $$
Differential of Stereographic Projection
$$ D_{\text{sproj}}(x) = \begin{bmatrix} \frac{1}{1-\sqrt{|K|}x_{n+1}} \mathbf{I} & \frac{\sqrt{|K|}}{(1-\sqrt{|K|}x_{n+1})^2} \begin{bmatrix} x_1\\ \vdots\\ x_n\\ \end{bmatrix} \end{bmatrix} $$
Mapping of Tangent Vectors
Putting everything together we get the mapping that maps tangent vectors on the hyperboloid to tangent vectors on the Poincaré ball.
$$ \mathbf{v} \mapsto D_{\text{sproj}}(x)\mathbf{v} $$
Is that correct?