2
$\begingroup$

For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following

i) The positive sheet of a two-sheeted hyperboloid (i.e. points that fulfill the equations $x^2 - y^2 - z^2 = 1, x>0$).

ii) This is in Minkowski space.

iii) The geodesic between two points in this model is the intersection of the hyperboloid with the plane that is determined by the two points and the origin (I assume that this would be the shortest path on the hyperboloid both in Euclidean and Minkowski space, can someone clarify?).

iv) The quadratic form in the model is given as $Q(x) = x_0^2 - x_1^2 - x_2^2$, which gives us the bilinear form $B(x,y) = x_0 y_0 - x_1 y_1 - x_2 y_2 $ which gives us the distance between two points $d(x,y) = \text{arccosh} (B(x,y))$.

I would like to know what parts of the above are a definition and what is a theorem. In Foundations of Hyperbolic Manifolds by Ratcliffe, the author begins by stating that the hyperboloid model is the points that fulfill $x^2 - y^2 - z^2 = 1, x>0$ in Lorentzian 3-space, so $R^3$ with a Lorentzian inner product $x \circ y = -x_1 y_1 + x_2 y_2 + x_3 y_3$, Lorentzian norm $||x|| = (x \circ x)^{1/2}$ and Lorentzian distance $d_L(x,y) = ||x-y||$. But then the author goes on to say that the model of hyperbolic $n$-space shold be the points that fulfill $||x||^2 = -1$ (the sphere of unit imaginary radius) and from there he gets the equation in i). So I assume that the only parts of the definition that we need for this model is that the hyperbolic plane is the points that fulfill $||x||^2 = -1$ in Lorentzian (Minkowski) 3-space? Is this a correct understanding?

$\endgroup$
2
$\begingroup$

Well the definition of the hyperbolic plane is not just a definition of a set. Indeed the hyperboloid model is diffeomorphic to a Euclidean plane. But not isometric!

So, (one of) the (right) definition is:

takes the component of the hyperboloid with $x>0$ and, as metric, restrict the Minkowski metric (that with signature ++-) (exercice: check that this is indeed positive definite)

As for theorems, item $iii)$ is now a theorem.

I suggest to look the great book by Benedetti and Petronio "Lectures on hyperbolic geometry" http://www.zbmath.org/?q=an:00107544

It is smaller, and perhaps less complete than Ratcliffe, but is very well written and readable. You will find there all the answer to your doubts.

$\endgroup$
  • 1
    $\begingroup$ Thanks a lot, I'll check out that book. So item iii) is a theorem and I suppose item iv) follows as well from the Minkowski metric? In other words, only i) and ii) are the definitions that we need? By the way, are Minkowski and Lorentizan space synonyms? $\endgroup$ – Sid Mar 15 '14 at 20:11
1
$\begingroup$

As a matter of taste I would avoid defining an object using coordinates. In this fashion you obtain the object plus a privileged set of coordinates, those that you used in the definition. So the simplest definition of $\mathbb{H}^n$ seems to be: Let $V$ be a $n$-dimensional vector space endowed with a bilinear form $g$ of Lorentzian signature $(-,+,\cdots,+)$ (notice that no basis is selected). We define $\mathbb{H}^n$ as a connected component of the locus $\{v\in V: g(v,v)=-1\}$ endowed with the metric induced from the bilinear form. All the other results are theorems. By the way, in my opinion, this should be regarded not as a model but as the definition of hyperbolic space. Then it turns out that it is simply connected, complete, homogeneous, and with sectional curvature -1 and all Riemannian manifolds with these properties are isometric, so the latter properties are often used in the definition of hyperbolic space. This point of view is not economical as it requires many results just to give a definition!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.