Revolution (Constant Curvature I)

One of the standard elementary surfaces is the Pseudosphere, a surface of revolution of constant negative curvature.



It can be parametrized using elementary function, and the profile curve is the so-called tractrix. Another elementary surface of constant negative curvature is Dini’s surface, where the tractrix is used to produce a helicoidal surface.



From here on, things get tricky. Other such surfaces of revolution require elliptic integrals. Here is the entire zoo (more or less):


Common to all examples is that they necessarily produce singularities. More precisely, there is no complete surface of constant negative curvature in Euclidean space. This is a famous theorem of Hilbert. At the core of the proofs I know is the behavior of the asymptotic lines


Above is the pseudosphere with one family of these asymptotic lines, drawn as ribbons. At the equator, they become horizontal. As the second family is the mirror image of this family, at the equator their tangent vectors become linearly dependent. This shows that while the asymptotic curves exist in the northern and southern hemipseudospheres, the surface itself is singular at the equator, because, alas, on negatively curved surfaces the asymptotic directions are linearly dependent. For the general surfaces of revolution, the asymptotic lines touch both singular latitudes. The image above looks odd because our brain wants to believe that curves on a surface meet at right angles. They don’t.

Pseudoasl2One of the key features of the asymptotic lines is that they form a Chebyshev net: Opposite edges of the net quadrilaterals have the same length. Thus you can stretch a loosely knitted square mesh over this surface to keep it warm. The standard proof of Hilbert’s theorem continues to show that any net parallelogram has area bounded above by some constant. However, a simply connected complete surface of constant negative curvature has necessarily infinite area, which leads to a contradiction. This was one of the earliest global results in differential geometry.