Hyperbolloid Model Translations

Question

Although the hyperboloid model of hyperbolic geometry has natural analogues of reflections and rotations, I am having trouble finding any linear transformation which is distance preserving and analogous to translation in Euclidean geometry. Recall that if $\textbf{v},\textbf{w} \in \mathbb{R}^n$, then our metric, $d(\cdot,\cdot)$, is derived from out inner product, the relation of which is given by $$\cosh(d(\textbf{v},\textbf{w}))=(v_1w_1-\sum_{i=2}^n v_iw_i)=\langle \textbf{v}, \textbf{w} \rangle$$ The question is simple: does there exist such a transformation and if so, what is it?

Answer 1

Use the isometry obtained from the $n \times n$ identity matrix by replacing the upper left $2 \times 2$ submatrix with $\pmatrix{5/4 & 3/4 \\ 3/4 & 5/4}$. This matrix preserves the line which is the intersection of the hyperboloid with the $x_1,x_2$-coordinate plane, and it translates along that line.

This matrix is the one which has a "light ray" eigenvector $\pmatrix{+1 \\ +1 \\ 0 \\ \vdots}$ with eigenvalue $2$, and a "light ray" eigenvector $\pmatrix{+1 \\ -1 \\ 0 \\ \vdots}$ with eigenvalue $1/2$.

That's the typical behavior of a hyperboloid model translation: there is one light ray eigenvector with eigenvalue $\lambda>1$, and another light ray eigenvector with eigenvalue $1/\lambda < 1$. The line preserved by this translation is the intersection of the hyperboloid model with the plane spanned by those two eigenvectors.