Poincare’s Half-plane Model (bis)

In a previous post, we considered Poincaré’s half-plane model for hyperbolic geometry in two dimensions. The half-plane model comprises the upper half plane ${H = \{(x,y): y>0\}}$ together with a metric

$\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.$

It is remarkable that the entire structure of the space follows from the metric.

In the earlier post, we derived the total curvature by evaluating the Riemann tensor. Here, we compute the curvature directly, using Gauss’s “Remarkable Theorem”.

The Principal Curvatures of a Surface

We define the principal curvatures ${k_1}$ and ${k_2}$ of a surface ${S}$ as the maximum and minimum values of the curvature of a curve formed by the intersection of ${S}$ with planes containing the normal to the surface. Clearly, ${k_1}$ and ${k_2}$ require measurements external to the surface itself; they are extrinsic quantities. In a remarkable theorem, the Theorema Egregium, Gauss showed that the total curvature ${\kappa=k_1k_2}$ can be evaluated intrinsically, using only quantities that can be measured within the surface itself.

Saddle surface with tangent plane and normal planes in directions of principal curvatures [Wikimedia Commons].

Saddle surface with tangent plane and normal planes in directions of principal curvatures [Image from Wikimedia Commons].

An Easy Way to Evaluate Curvature

We assume that the surface is specified in terms of two parameters ${u}$ and ${v}$ as

$\displaystyle x = x(u,v) \,, \qquad y = y(u,v) \,, \qquad z = z(u,v)\,.$

or $\mathbf{r}=\mathbf{r}(u,v).$ The First Fundamental Quadratic Form is

$\displaystyle ds^2 = E du^2 + 2F du\,dv + G dv^2.$

where the functions ${E(u,v)=\mathbf{r}_u\mathbf{\cdot r}_u}$ , ${F(u,v)=\mathbf{r}_u\mathbf{\cdot r}_v}$ and ${G(u,v)=\mathbf{r}_v\mathbf{\cdot r}_v}$ can be evaluated within the surface. Then the total curvature may be expressed in terms of these three functions and their first and second derivatives. In the case of orthogonal coordinates, ${F=0}$ , we get:

$\displaystyle \kappa = -\frac{1}{EG}\left[ \frac{1}{2}(E_{vv}+G_{uu}) - \frac{1}{4}\left( \frac{E_uG_u+E_v^2}{E} + \frac{E_vG_v+G_u^2}{G}\right) \right]$

(see [1], pg. 183). For the Poincaré half-plane, the First Fundamental Quadratic Form is

$\displaystyle ds^2 = \frac{du^2+dv^2}{v^2}$

so we have ${E = G = 1/v^2}$ and ${F=0}$ . Then the expression for ${\kappa}$ can be evaluated and yields the result:

$\displaystyle \kappa= -1 \,.$

This is simpler and more direct than evaluating the entire fourth-order Riemann tensor.

Coming Soon

In the next post [ Poincaré’s Half-plane Model (ter) ] we will look at some geodesic triangles in the hyperbolic half-plane and see how the angle deficit is related to the area. This will introduce the key topic of the Gauss-Bonnet Theorem.

Sources

[1] Lanczos, Cornelius, 1979: Space through the ages: The evolution of geometric ideas from Pythagoras to Hilbert and Einstein. Academic Press. ISBN 0124358500.

[2] Lynch, P, 2013: Curvature of the Poincaré Half-Plane. ( PDF )

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