Hyperboloid model

Example

Let $n \ge 1$ and give $\R^{n+1}$, with coordinates $x = (x_0, x_1, \ldots, x_n)$, the symmetric bilinear form $$B(x, y) = x_0 y_0 - x_1 y_1 - \cdots - x_n y_n.$$ The hyperboloid model of hyperbolic $n$-space is the upper sheet of a hyperboloid of two sheets, $$H^n = \set{x \in \R^{n+1} : B(x, x) = 1, \ x_0 > 0},$$ with the distance $d(x, y) = \operatorname{arcosh} B(x, y)$. On each tangent space of $H^n$ the form $-B$ is positive definite, and the resulting Riemannian manifold has constant sectional curvature $-1$. Its geodesics, the lines of hyperbolic geometry, are the nonempty intersections of $H^n$ with 2-dimensional linear subspaces of $\R^{n+1}$.

Check for $n = 1$: the curve $\gamma(t) = (\cosh t, \sinh t)$ lies on $H^1$, since $\cosh^2 t - \sinh^2 t = 1$ and $\cosh t > 0$. Its velocity is $\gamma'(t) = (\sinh t, \cosh t)$, and $-B(\gamma', \gamma') = \cosh^2 t - \sinh^2 t = 1$, so $t$ is arc length. The formula gives $d(\gamma(0), \gamma(t)) = \operatorname{arcosh}(\cosh t) = \abs{t}$, in agreement.