11.6: Curvature
In a letter from 1824 Gauss writes:
The assumption that the sum of the three angles is less than π leads to a curious geometry, quite different from ours but completely consistent, which I have developed to my en- tire satisfaction, so that I can solve every problem in it with the exception of a determination of a constant, which cannot be designated a priori. The greater one takes this constant, the nearer one comes to Euclidean geometry, and when it is chosen indefinitely large the two coincide. The theorems of this geometry appear to be paradoxical and, to the uninitiated, absurd; but calm, steady reflection reveals that they contain nothing at all impossible. For example, the three angles of a triangle become as small as one wishes, if only the sides are taken large enuf; yet the area of the triangle can never exceed a definite limit, regardless how great the sides are taken, nor indeed can it ever reach it.
In modern terminology, the constant that Gauss mentions, can be expressed as \(1/\sqrt{-k}\), where \(k \le 0\), is the so-called curvature of the neutral plane, which we are about to introduce.
The identity in Exercise 11.4.1 suggests that the defect of a triangle should be proportional to its area. (The area in the neutral plane is discussed briefly in the end of Chapter 20, but the reader could also refer to an intuitive understanding of area measurement.)
In fact, for any neutral plane, there is a nonpositive real number \(k\) such that
\(k \cdot \text{area} (\triangle ABC) + \text{defect} (\triangle ABC) = 0\)
for any \(\triangle ABC\). This number \(k\) is called the curvature of the plane.
For example, by Theorem 7.4.1 , the Euclidean plane has zero curvature.
By Theorem 11.3.1 , the curvature of any neutral plane is nonpositive.
It turns out that up to isometry, the neutral plane is characterized by its curvature; that is, two neutral planes are isometric if and only if they have the same curvature.
In the next chapter, we will construct a hyperbolic plane; this is, an example of neutral plane with curvature \(k = -1\).
Any neutral planes, distinct from Euclidean, can be obtained by scaling the metric on the hyperbolic plane. Indeed, if we scale the metric by a positive factor \(c\), the area changes by factor \(c^2\), while the defect stays the same. Therefore, taking \(c = \sqrt{-k}\), we can get the neutral plane of the given curvature \(k < 0\). In other words, all the non-Euclidean neutral planes become identical if we use \(r = 1/\sqrt{-k}\) as the unit of length.
In Chapter 16, we discuss spherical geometry. Altho spheres are not neutral planes, the spherical geometry is a close relative of Euclidean and hyperbolic geometries.
Nondegenerate spherical triangles have negative defects. Moreover, if \(R\) is the radius of the sphere, then
\(\dfrac{1}{R^2} \cdot \text{area} (\triangle ABC) + \text{defect} (\triangle ABC) = 0\)
for any spherical triangle \(ABC\). In other words, the sphere of the radius \(R\) has the curvature \(k = \dfrac{1}{R^2}\).