The Poincare Half-plane

Intrinsic geometry also means that we can define and study abstract surfaces that cannot be embedded in 3-dimensional space; or similarly, that we can determine the intrinsic geometric properties of space-time without having ßpace-time" embedded in a larger space.  

For example, we can define a new geometry on the plane by giving it a non-Euclidean fundamental form. How would we know that it was truly different? This is exactly what Henri' Poincare' did when he introduced the fundamental form

ds2 =  du2+dv2 v2

(6)

to the upper half of the uv-plane. The result is called the Poincare half-plane and is a model of hyperbolic geometry.

If we use (6) to measure distances, then the geodesics are the vertical lines and semicircles parameterized by

u = Rtanh(t) +p,  v = R sech( t) ,   t in ( -¥,¥)

for R and p constant. For example, because distances become shorter as v increases under the Poincare metric (6), the distance from ( -1,1) to ( 1,1) along a semi-circle of radius Ö2 centered at the origin is 1.7627, which is shorter than the distance of 2 from ( -1,1) to (1,1) along the line v = 1.

Thus, vertical lines and semi-circles centered on the u-axis are the straight lines" in the Poincare half-plane. Through a point P not on a semi-circle, there are infinitely many other semi-circles centered on the x-axis that pass through P. Thus, in the Poincare half plane, there are infinitely many parallel lines to a given line l through a point P not on l.

Finally, we can use the Theorem Egregium to calculuate the curvature of the Poincare half-plane.

At the end of this section, we have a very accessible proof of the Theorem Egregium. However, before proving the theorem, let's examine its value as a means of exploring the geometry of a surface given only its fundamental form.       

EXAMPLE 5    What is the curvature of the hyperbolic plane, which is the upper half plane with the Poincare fundamental form

ds2 =  du2+dv2 v2

Solution: Since ds² = v^-2du²+v^-2dv², the metric coefficients are g₁₁ = g₂₂ = v^-2. Thus, g = g₁₁g₂₂ = v^-4.  and since the metric is conformal, we have

K =  -1 2 v-4 é ê ë  ¶ ¶v æ ç è  -2v-3 v-4 ö ÷ ø +  ¶ ¶u æ ç è  0 v-4 ö ÷ ø ù ú û =  -1 2v-2 é ë  ¶ ¶v (-2v-1) +  ¶ ¶u ( 0) ù û =  -1 2v-2 ( 2v-2) = -1

Thus, the curvature of the hyperbolic plane is K = -1. That is, the hyperbolic plane is a surface of constant negative curvature, and as a result, it cannot be studied as a surface in ordinary 3 dimensional space. Instead, all information about the hyperbolic plane must come from the intrinsic properties derived from its fundamental form.