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For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the angle between the two vectors).

Is there an analogous coordinate representation for n-dimensional hyperbolic space? And with a similarly simple distance metric? I'm expecting something like an $n$- or $(n+1)$-tuple, where the distance function doesn't treat any particular function specially. In fact it would be great if there were a parameter $k$ for curvature to a distance function that would differentiate between spherical, euclidean and hyperbolic and work for any dimension. But if there are others for n-dimensional, that would be great.

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2 Answers 2

I imagine what you want is the half-hyperboloid model in $\mathbf R^{n+1}$ with the Lorentz metric. So, the set is $x_{n+1} >0$ in $$ x_1^2 + x_2^2 + \ldots + x_n^2 - x_{n+1}^2 = -1.$$ There is a useful distinguished North Pole at $(0,0,0,\ldots,0,1).$ Geodesics, at unit speed, going through the North Pole are rotations of $$(\sinh t,0,0,\ldots,0,\cosh t). $$ Indeed, all isometries preserve the ambient metric. Let's see, all geodesics are the intersection of a 2-plane passing through the origin with the hyperboloid. As a result, there is a isometry to the Beltrami-Klein model given by central projection around the origin, to the $n$-ball $$x_{n+1} =1, \; \; x_1^2 + x_2^2 + \ldots + x_n^2 < 1. $$ Finally, an isometry of Beltrami-Klein to the Poincare $n$-ball model is given by vertical projection to the lower hemisphere of the ordinary $n$-sphere in $\mathbf R^{n+1}$ followed by stereographic projection around the North Pole to the Poincare $n$-ball $$x_{n+1} =0, \; \; x_1^2 + x_2^2 + \ldots + x_n^2 < 1. $$ The diagram for some of these maps is page 255, figure 246 of Hilbert and Cohn-Vossen, Geometry and the Imagination. However, Hilbert makes the disk containing the final Poincare model have radius 2, I have altered things a little to get radius 1.

This ought all to be in Spivak's five-volume book.

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That's quite pretty a figure... –  J. M. Oct 2 '11 at 21:19

There's nothing quite as simple as your example for hyperbolic space, because the example is based on embedding the $n$-sphere in Euclidean $n+1$-space, and no such smooth isometric embedding is possible for hyperbolic space.

The closest to what you're asking might be to use $n$ coordinates $x, y, \ldots, z$ with a metric given by $$d\sigma^2 = \frac{dx^2+dy^2+\cdots+dz^2}{(1+\frac{k}{4}(x^2+y^2+\cdots+z^2))^2}$$ where varying $k$ can produce either Euclidean space for $k=0$, an $n$-sphere minus one point (via stereographic projection) for $k>0$, and hyperbolic space for $k<0$ and $x^2+y^2+\cdots+z^2<4/k$ (which produces the $n$-dimensional analogue of the Poincaré disk model). In each cases, the denominator depends only on which point we're at, and the tangent space at any point is a simple uniform scaling away from the coordinate tangent space.

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