Poincare's Vision and Semantic Unfolding in Geometry: The Natural Intrinsic Geometry of Riemann Surfaces

In the theory of Riemann surfaces one of the most fundamental results is the Uniformization theorem, according to which, any simply connected Riemann surface is conformally equivalent to the Riemann sphere, or to the complex plane, or to the unit disk. The Uniformization theorem explicates the geometric semantics incorporated into the complete unfolding of a multiply-extended variable magnitude, which is obtained by the Riemannian strategy of cutting and amalgamating into the emergent universal covering simply-connected Riemann surface of this function, where it becomes uniformly valued.

According to the Uniformization theorem, the type of uniformization applicable to a topological surface bearing holes in order to obtain its universal covering Riemann surface depends on the genus of this surface. More precisely, those of genus zero are their own universal covering Riemann surfaces, isomorphic to the Riemann sphere; those of genus one are universally covered by the complex plane; and those of genus at least two are universally covered by the unit disk.

Poincare's crucial contribution to the geometry of Riemann surfaces in relation to the connective thread of all the relevant ideas surrounding the issue of uniformization may be accurately described from our semantic unfolding perspective as follows: Given a regular polygon in the Euclidean plane (identified with the complex plane), or the hyperbolic disc, with particular conditions, Poincare viewed the pertinent notion of group action from three different unifying perspectives, namely as an isometry group of geometric transformations that induces a tessellation on a simply-connected universal covering space, as the Galois group of covering transformations of a universal covering space, and as the fundamental group of a quotient space of that universal covering space.

Poincare's Disk Hyperbolic Ontophainetic Platform

Escher's Circle Limit IV

Constellatory Unfolding of the Genus

Constellatory Unfolding of the Double Torus: The Hyperbolic Octagon

Poincare's Phraseology

Poincare in the third chapter of his "Science and Hypothesis'' describes the unit disk model of the hyperbolic plane as follows:

Let us assume ... a world enclosed in a large circle and subject to the following laws: The temperature in this world is not uniform; it is largest at the centre, and it diminishes as one moves away from the centre, so that it reduces to absolute zero when one reaches the circle where this world is enclosed.

I will moreover specify the following law by which this temperature varies. Let R be the radius of the limit circle; let r be the distance from the point under consideration to the centre of this circle. The absolute temperature will be proportional to R^2-r^2.

I will additionally assume that, in this world, all bodies have the same coefficient of dilatation, in such a way that the length of any ruler shall be proportional to its absolute temperature. Finally, I will assume that an object transported from one point to another, whose temperature is different, shall immediately reach thermal equilibrium with its new location.

Nothing in these hypotheses is contradictory or unimaginable. A moving object will then become smaller and smaller as it approaches the limit circle. Let us first observe that, if this world is finite from the point of view of our customary geometry, it will appear infinite to its inhabitants. In fact, when they wish to approach the limit circle, they will get colder and become smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the limit circle.

If, for us, geometry is merely the study of the laws by which non-deformable solids may move; for these imaginary beings, it will be the study of the laws that drive solids deformed by these differences in temperature about which I have just spoken...For brevity, I shall, with the reader's permission, call such a movement a non-Euclidean displacement.

Thus beings like us, whose education would take place in such a world, would not have the same geometry as us. If these imaginary beings founded a geometry, ... it would be non-Euclidean geometry.