Curvature: True Versus Apparent

by Jonathan Tennenbaum

In the course of our discussion of “ the first measurement of the Universe” the concept of curvature arose at first in a {negative way}: the {impossibility} of representing the visible arrangement of stars in the heavens on a flat surface. Any attempt to create such a star map inevitably distorts the constellations and the angular relations between the constellations; and the distortion becomes ever greater, the larger the portion of the heavens we attempt to map. Our study of the characteristic singularities connected with this mapping problem, led us to the regular solids. With the discovery of those solids, the concept of curvature, at first a purely negative one, took on a definite form.

Now the concept of curvature, so developed, is something entirely different from the idea of “curvedness” associated with our sense-perception. Unlike the latter, true curvature involves an ontological singularity and can be grasped only by the cognitive powers of the mind. Carl Gauss’ 1827 “General investigations of curved surfaces” focussed on that crucial difference. Taking the case of simple geometrical surfaces as his pedagogical starting-point, Gauss developed the concept of so-called intrinsic or internal curvature of a manifold as an analysis-situs notion, independent of the manner in which such a manifold might happen to be represented in visual or other formal terms.

The significance of this problem should be clear enough. For example: How can how can we overcome the “spin” which our naive sense-perception tends to impose upon any portion of physical reality? How do we distinguish the true characteristics of a process, from those which merely reflect the effect of arbitrary, extraneous assumptions and other distorting factors dragged in “from the outside”? Gauss’ work on the orbit of Ceres, his work on geodesy and his collaboration with Wilhelm Weber on so-called absolute electrodynamic measurements, all depend upon his approach to this critical issue.

To start off with a very simple illustration, compare the following three surfaces: 1) a flat, plane surface 2) the surface of a cylinder 3) the surface of a sphere. Here is the question: We have begun to elaborate a concept of (non-zero) curvature as a characteristic which absolutely distinguishes the spherical surface from the flat one. Now, what should we say about the cylinder? As a form in visible space, the cylindrical surface certainly seems to possess a curvature. But according to the conception adopted by Gauss, the internal or intrinsic curvature of the cylindrical surface is {zero}: it is essentially flat and indistinguishable from the plane surface “in the small”! A paradox? Let us look into the matter more closely.

Gauss related his notion of internal geometry and internal curvature to the characteristics of the minimum pathways (geodesics) in the given manifold. In the plane, these turn out to be straight lines, while on a spherical surface they are portions of great circles. What are they on the cylindrical surface?

Take a smooth, rigid cylinder (preferably from wood and not too small diameter), cut a rectangle out of smooth-surfaced paper so that it wraps once around the cylinder, and fix it tightly to cylinder by tape or tacks. Now take a piece of thread and stretch it tight along the surface between any two points. This defines a geodesic or “shortest curve” on the surface.

In two special cases the form of these lines is obvious: if the direction between the two points is parallel to the cylinder’s axis, then the shortest curve connecting them on the surface is a straight line; while if the two points are at the same “height” along the cylindrical axis, the geodesic is an arc of the circle which results from cutting the cylinder perpendicular to the axis at the height of the two points. In other cases, however, the form of the geodesic appears more complicated.

To get an overview of the geodesics at any given point, construct a “geodesic circle” as follows. Fix any position on the paper-covered cylinder by a pin or nail. Tie one end of a piece of thread around the nail and the other around the tip of a pen or pencil, and stretch it tight along the surface (or alternatively, use a loop of thread). Trace the curve which results from moving the tip of the pen on the cylindrical surface with the thread kept tight. Also, trace the form of the thread on the cylinder at several positions during that process.

Now, what happens to these curves if we unwrap the paper from the cylinder and lay it flat? The crucial observation to be made is, that the lengths of the curves, traced on the paper, are not sensibly changed by the unwrapping process! In consequence, the minimal curves on the cylinder become minimal curves on the flattened surface — i.e. straight lines –, and the geodesic circle on the cylinder becomes an ordinary plane circle.

What does this mean? Insofar as the internal metrical properties of the surface are embodied in the array of minimum pathways on that surface, there is no discernible difference between the cylindrical surface and the flat surface which results from unwrapping it. In terms of internal geometry, the flattened surface is a perfect map of the cylindrical original. Hence, the cylindrical surface must have the same internal curvature as the flat one — zero curvature!

Of course, we should not completely overlook two points: Taken as a whole, the complete cylindrical surface contains {closed cycles} — e.g. a circle enclosing the axis — which are not present in the flat surface. To unwrap the cylindrical surface, we must cut or tear it lengthwise, introducing a discontinuity. So the apparent equivalence applies only to smaller, local regions of the cylindrical surface, which do not wrap fully around the axis. Secondly, an “infinitely thin” mathematical surface, completely unchanged by the unwrapping process, does not exist in the physical universe in a literal sense — any more does a {purely} internal metric, which would not react {in some way} to a change in the relationship between the given manifold and the rest of the Universe.

These points, however, in no way obviate the methodological issue Gauss is addressing. We are rather impelled to pose once more the question which originally launched our whole investigation, but in somewhat different terms than before:

How could we {know}, by measurements taken entirely “inside” a spherical surface (or in any arbitrarily small portion of that surface) — i.e. by measurements made without explicit reference to the sphere’s apparent form in visual space — that no flattening-out of a spherical manifold is possible?