THE BEGINNINGS OF DIFFERENTIAL GEOMETRY
Fifty-two years after Gauss’ 1799 doctoral dissertation on the fundamental theorem of algebra, his student, Bernhard Riemann, submitted, to Gauss, an equally revolutionary doctoral dissertation that took Gauss’ initial discovery into a new, higher, domain. Riemann’s thesis, “Foundations for a general theory of functions of a single variable complex magnitude”, built on the foundations of Gauss’ own work, established a complete generalization of the principles of physical differential geometry that was set into motion by Kepler nearly 250 years earlier.
It is beneficial, and perhaps essential, as a preliminary to a more detailed discussion of Riemann’s work itself, to review three exemplary discoveries of physical principles, that taken together, trace the historical development of the ideas leading into Riemann’s work: Kepler’s principles of planetary motion; the Leibniz-Bernoulli discovery of the principle of the catenary; and Gauss’ own work in geodesy. All three, while seemingly diverse, are in fact intimately connected. They all deal, in one way or another, with investigations into the nature of universal gravitation, and, taken together, they comprise a succession of concepts of increasing generality and power.
Begin first with Kepler. Taken in its entirety, from the Mysterium Cosmographicum to the Harmonice Mundi, Kepler’s work demonstrates that the action governing any planet at any moment is a function of the principle that organizes the solar system as a whole; the principle of universal gravitation. Kepler discovered that this principle has an harmonic characteristic, which determines that the planetary orbits are elliptical, not circular. The unique shape of each individual elliptical orbit is determined, not by each planet alone, nor by the pair-wise interaction of that planet with the Sun, but by the harmonic relationship among the maximum and minimum speeds of all the planets. In other words, the action of the planet at any moment is determined by these extremes, between which, the planet’s orbit “hangs”. The magnitudes of these “hanging points”, are not arbitrary, but when taken all together, conform, approximately, to the harmonic ordering of the musical scale.
The eccentricity of the planetary orbits posed a challenge to Kepler because he had no mathematical means to determine the exact position, direction and velocity of each planet at every moment, so he demanded the invention of a new mathematics. Kepler prescribed that such a mathematics must be able to determine how the harmonic principle that determines the planet’s extremes, is expressed, throughout the entire orbit, and he took the first steps toward developing that mathematics. (See Riemann for Anti-Dummies Parts 1-6)
Responding to Kepler’s demand, Leibniz and his collaborator, Johann Bernoulli developed the calculus, the most general expression of which is demonstrated by their joint effort on the catenary. At first glance, the catenary appears similar, in principle, to a planetary orbit, in that the shape of the curve seems to be determined by the position of the points from which chain hangs. As the position of these “hanging points” changes, the chain re-orients itself, so that its overall shape is maintained. In this respect, the relationship of these hanging points to all the other points on the catenary, initially seems analogous to the relationship between the extreme speeds of a planet to the entire orbit. But, as Bernoulli showed in his book on the integral calculus, all points on the catenary, except the lowest point, are, at all times, hanging points. (The reader should review Riemann for Anti-Dummies Parts 10 “Justice for the Catenary”, and chapter 4 of “How Gauss Determined the Orbit of Ceres”, to perform the experiments indicated therein.(fn. 1.)) This is, in fact, an inversion of the principle expressed in Keplerian orbits. In the case of the planet, the orbit, “hangs” between its two extremes. For the catenary, the extreme, that is the lowest point, is the one point that does no hanging. (In Cusa’s terms it is the point that is simultaneously motion and no-motion.) Applying Leibniz’ calculus, Bernoulli demonstrated how the catenary is “unfolded” from this lowest point. (fn. 2.)
Leibniz, in turn, demonstrated that this physical principle also reflected the characteristic exhibited by the logarithmic (exponential) function. (See Leibniz paper on catenary.) Thus, the hanging chain is characterized by the same transcendental principle that subsumes the generation of the so-called algebraic powers, and which is exhibited in other physical processes such as biological growth and the musical scale, as well. Consequently, the characteristics of the logarithmic (exponential) function, is an expression of a physical principle, not a mathematical one.
Now, compare the above described examples with Gauss’ discovery of the Geoid. From 1818 to 1832 Gauss carried out a geodetic survey of the Kingdom of Hannover. This involved determining the physical distances along the surface of the Earth by laying out triangles and measuring the angles formed by the “line of sight” sides. The paradox Gauss confronted was that the relationship between the lengths of the sides of the triangles and the angles, is a function of the shape of the Earth. (fn.3.) However, the shape of the Earth could not be known in advance of the measurements. The problem was further complicated by the fact that all the measurements were taken with respect to the direction of the pull of gravity, as determined by the direction of a hanging plumb bob. Like the relationship between the angles of a triangle and the lengths of the sides, the direction of the pull of gravity depends on the shape of the Earth. For example, if the Earth were spherical, the plumb bob would always point toward the center of the Earth. If the Earth were ellipsoidal, the plumb bob would point to different places, depending on where on the ellipsoid the measurement was being taken. Gauss showed that the problem was even more complicated, because the Earth’s shape was very irregular. (See Riemann for Anti-Dummies Part 17.)
Here Gauss was confronted with exactly the same type of problem as Kepler and Leibniz before him. Existing mathematics could not measure such an irregular shape. All previous approaches began with an a priori assumption of the shape of the Earth, one that conformed to existing mathematical knowledge. (This brings to mind Gallileo’s foolish insistence that the catenary was a parabola because that was the shape in the mathematical textbooks which looked most like a catenary. The chain, however, did not read Gallileo’s preferred texts.) Gauss abandoned all such attempts to fit the Earth into an assumed shape, declaring that the geometrical shape of the Earth is that shape that is everywhere perpendicular to the pull of gravity. In other words, instead of assuming an imaginary shape, and measuring the real Earth as a deviation from the imaginary one, Gauss rejected the fantasy world altogether. (Something more and more people should want to do these days as the global monetary systems disintegrates.) The physically determined shape that Gauss measured has since become known as the Geoid.
While the Geoid is an irregular surface, its irregularity is “tuned” so to speak by the motion of the Earth on its axis. Like the planetary orbit, or the hanging chain, that motion determines the positions of two, “hanging points”, specifically the north and south pole, from which the Geoid hangs.
However, since the Geoid is a surface, it has a different relationship to its poles, than the planetary orbit to its extremes, or the catenary to the lowest point. The latter two cases express the relationship between singularities and action on a curve. The former expresses the relationship between singularities and action on a surface, from which the action along the curves is derived.
The problem Gauss confronted was that since the physical triangles he measured on the surface of the Geoid were irregular, how could the lengths of the sides be determined from the angles, without first knowing the relationship between the lengths and the angles, i.e., the shape of the surface? To solve the problem, Gauss recognized that since all his measurements were angles, he could free himself from having to assume the Earth’s shape before he could determine his measurements, if he could project these angles from one surface to another, for example, from the geoid, to an ellipsoid, to a sphere and back again. Like Kepler and Leibniz, Gauss could not do this within the existing mathematics. So he invented a new one.
Gauss described the beginnings of this new mathematics in several locations, most notably his 1822 memoir on the subject of conformal mapping, that was awarded a prize from the Royal Society of Sciences of Copenhagen. Riemann relied heavily on this paper for the foundations of his own doctoral dissertation.
Conformal mapping is a term, invented by Gauss, to refer to transformations from one surface to another in which the angles between any curves on that surfaces are preserved. In his memoir, Gauss described conformal mapping as a transformation where, “the lengths of all indefinitely short lines extending from a point in the second surface and contained therein shall be proportional to the lengths of the corresponding lines in the first surface, and secondly, that every angle made between these intersecting lines in the first surface shall be equal to the angle between the corresponding lines in the second surface.”
To get an idea of what this means, perform the following experiment. Take a clear plastic hemisphere and draw a spherical triangle on it with heavy black lines. Go into a dark room and, using a flashlight, project the triangle onto the wall. If you hold the flashlight at the center of the hemisphere, the curved lines of the spherical triangle will be transformed into straight lines. If you then move the flashlight from the center of the hemisphere to a pole, the projected straight lines will become curved again, and the angles between them will be equal to the angles between the sides of the original triangle on the hemisphere.
To discover experimentally the difference between these two projections, tape cardboard circles of differing sizes onto the plastic hemisphere. (The circles should vary from quite large to quite small.) Now perform the same projection with the flashlight as before. When the flashlight is at the center of the hemisphere, these circles project to ellipses. When the flashlight is at the pole of the hemisphere, the circles become more circular, with the smaller circles become more circular than the larger ones. In the first case, the transformation of the circles into ellipses indicates that the proportion by which figures are transformed changes depending on the direction of the transformation with respect to the poles. The second case shows that the transformations are proportional in all directions.
Thus, the conformal mapping of one surface to another involves a change in rotation and direction. Having done the work on Gauss’ fundamental theorem of algebra, you should be able to recognize, as Gauss did, that this type of change could only be represented in the complex domain, which is where we will begin next time.
1. Any two points on opposite sides of the lowest point hold up the weight of the chain hanging between them. The force required to hold up this weight is proportional to the sines of the angles made by the tangents to the catenary at this point, and a vertical line rising from the point at which the tangents intersect.
2. The reader is urged to preform the experiment described in the indicated NF article. Take a string and tie a weight in the middle of it. Hold the ends of the string in each hand and let the weight hang between them. As you move your hands apart, the tension you feel on your hands will increase. If you begin with your hands close together, the tension is relatively small. As you pull your hands apart, the tension increases, slowly at first, but the rate of increase in the tension grows, the farther apart your hands are to one another. Now try to move your hands apart, while the string slides between your fingers, so that the string on one side remains horizontal. The other hand will move in the shape of the catenary.
3. The reader can grasp this by comparing triangles drawn on a piece of paper, a sphere and an irregular shaped surface, such as a watermelon.