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In hyperbolic geometry, the metric is often defined as $$ds=\frac{\sqrt{dx^2+dy^2}}{y}$$ Where did this metric come from? I have thought long and hard about this question, but have no satisfactory answer.

algebraically_speaking
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  • the first one was Beltrami. People usually do theory of curves and surfaces in $\mathbb R^2$ and $\mathbb R^3$ before attempting Riemannian Geometry. – Will Jagy Nov 08 '14 at 17:50
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    I added a bounty with the hope of getting an answer to this question along the following lines: is there a (system of) differential equations, that a metric on a space must satisfy? In that way this hyperbolic metric is arrived at by solving this DE? --- Is there anyway to come up with this metric that completely lacks mystery? Cheers, – User0112358 May 22 '17 at 11:59
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    @User0112358: If you want a Riemannian metric invariant under $PSL(2,R)$, then this is the only possible metric (up to a scale). Why would you want to have such a group of isometries? Because you want a conformal action transitive on the unit tangent bundle (so that SAS-congruence test would be true). – Moishe Kohan May 24 '17 at 02:32

3 Answers3

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This is the metric that results in the Poincaré half-plane model of the hyperbolic plane.

hmakholm left over Monica
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  • Are $dx,dy$ and $ds$ infinitesimal distances? Why would you have infinitesimal distances for a point? – algebraically_speaking Nov 08 '14 at 17:10
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    @algebraically_speaking: Sorry, it looked like you already knew what a Riemannian metric is in general. The basic property is that it can be used to define the length of a curve in the hyperbolic plane represented by $t\mapsto(x(t),y(t))$, $t\in[a,b]$ as $$\int_a^b \frac{1}{y(t)}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2},dt $$ – hmakholm left over Monica Nov 08 '14 at 17:20
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    @algebraically_speaking: You might also want to look in any calculus book regarding the path length integral of a curve in the Euclidean plane given by $t \mapsto (x(t),y(t))$, $t \in [a,b]$. That integral has the form $\int_a^b \sqrt{\bigl(\frac{dx}{dt}\bigr)^2 + \bigl(\frac{dy}{dt}\bigr)^2} , dt$. – Lee Mosher Nov 08 '14 at 17:51
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Notation is always troubling. The concrete calculations that go with the metric definition give what are called "geodesics," which are the replacement for straight lines, as they are the shortest curves between two points. in this case, there are just two types of geodesics, here parametrized by arc-length, vertical rays $$ (A, e^t) $$ and semicircles with center on the $x$-axis $$ (A + B \tanh t, \; \; B \operatorname{sech} t ) $$ with real constants $A$ of any sign and then $B>0.$

Will Jagy
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As you know, the fifth euclidean postulate says: Given a straight line $L$ in the plane and a point $p$ not in $L,$ there is exactly one straight line $S$ parallel to $L$ and passing by $p.$ Changing this postulate to "there are at least two lines such that..." leads to the so-called hyperbolic geometry. The upper-half space, $\mathbb{R}^{2}_+,$ equipped with that abstract fundamental form, or metric, is one example of a hyperbolic space, where this new "fifth postulate" holds. So, that metric comes from the attempt to construct a space with such modification of the euclidean postulate.

Qwertuy
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