Differential Geometry - Special Topics - A brief course - The history of mathematics

The history of mathematics: A brief course (2013)

Part VII. Special Topics

Chapter 39. Differential Geometry

Differential geometry is the study of curves and surfaces (from 1852 on, manifolds) using the methods of differential calculus, such as derivatives and local series expansions. This history falls into natural periods defined by the primary subject matter: first, the tangents and curvatures of plane curves; second, the same properties for surfaces and curves in three-dimensional space; third, minimal surfaces and geodesics on surfaces; fourth, the application (conformal mapping) of surfaces on one another; fifth, extensions of all these topics to n-dimensional manifolds and global properties instead of local.

39.1 Plane Curves

Besides the study of tangent and normal lines to plane curves, which was begun in connection with analytic geometry, certain auxiliary curves were studied—in particular the involute, which is defined below. Measures of curvature, such as the osculating circle (the circle that fits a curve up to second order near a point) and the radius of curvature became a focus of attention. As mentioned in Chapter 34, Brook Taylor used the assumption that the restorative force on a stretched string is proportional to its curvature in order to study the vibrations of strings. This assumption showed very good intuition, since the curvature is the second derivative with respect to arc length; under the approximations used to get a linear model for this phenomenon from Newton's laws of motion, that assumption yields precisely the correct equation.

39.1.1 Huygens

Struik (1933) and Coolidge (1940, p. 319) agree that credit for the first exploration of secondary curves generated by a plane curve—the involute and evolute—occurred in Christiaan Huygens' work Horologium oscillatorium (Of Pendulum Clocks) in 1673, even though calculus had not yet been developed. The involute of a curve is the path followed by the endpoint of a taut string being wound onto the curve or unwound from it. Huygens did not give it a name; he simply called it the “line [curve] described by evolution.” There are as many involutes as there are points on the curve to begin or end the winding process.

A cycloidal pendulum clock, fromHuygens' Horologium oscillatorium. Copyright © Stock Montage.

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Huygens was seeking a truly synchronous pendulum clock, and he needed a pendulum that would have the same period of oscillation no matter how great the amplitude of the oscillation was.1 Huygens found the mathematically ideal solution of the problem in two properties of the cycloid. First, a frictionless particle requires the same time to slide to the bottom of a cycloid no matter where it begins (the tautochrone property); second, the involute of a cycloid is another cycloid. He therefore designed a pendulum clock in which the pendulum bob was attached to a flexible leather strap that is confined between two inverted cycloidal arcs. The pendulum is thereby forced to fall along the involute of a cycloid and hence to trace another cycloid. Reality being more complicated than our dreams, however, this apparatus—like the mechanical drawing methods of Albrecht Dürer discussed in Chapter 31—does not really work any better than a circular pendulum.2

39.1.2 Newton

In his Fluxions, which was first published in 1736 after his death, even though it appears to have been written in 1671, Newton found the circle that best fits a curve. Struik (1933, 19, p. 99) doubted that this material was really in the 1671 manuscript. Be that as it may, the topic occurs as Problem 5 in the Fluxions: At any given Point of a given Curve, to find the Quantity of Curvature. Newton needed to find a circle tangent to the curve at a given point, which meant finding its center. However, Newton wanted not just any tangent circle. He assumed that if a circle was tangent to a curve at a point and “no other circle can be interscribed in the angles of contact near that point,. . .that circle will be of the same curvature as the curve is of, in that point of contact.” In this connection he introduced terms center of curvature and radius of curvature still used today. His construction is shown in Fig. 39.1, in which one unnecessary letter has been removed and the figure has been rotated through a right angle to make it fit the page. The weak point of Newton's argument was his claim that, “If CD be conceived to move, while it insists [remains] perpendicularly on the Curve, that point of it C (if you except the motion of approaching to or receding from the Point of Insistence C,) will be least moved, but will be as it were the Center of Motion.” Huygens had had this same problem with clarity. Where Huygens had referred to points that can be treated as coincident, Newton used the phrase will be as it were. One can see why infinitesimal reasoning was subject to so much doubt at the time.

Figure 39.1 Newton's construction of the radius of curvature, from his posthumously published Fluxions.

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Newton also treated the problem of the cycloidal pendulum in his Principia Mathematica, published in 1687. Huygens had found the evolute of a complete arch of a cyloid. That is, the complete arch is the involute of the portion of two half-arches starting at the halfway point on the arch. In Proposition 50, Problem 33 of Book 1, Newton found the evolute for an arbitrary piece of the arch, which was a much more complicated problem. It was, however, once again a cycloid. This evolute made it possible to limit the oscillations of a cycloidal pendulum by putting a complete cycloidal frame in place to stop the pendulum when the thread was completely wound around the evolute.

39.1.3 Leibniz

Leibniz’ contributions to differential geometry began in 1684, when he gave the rules for handling what we now call differentials. His notation is essentially the one we use today. He regarded x and x + dx as infinitely near values of x and img and dv as the corresponding infinitely near values of img on a curve defined by an equation relating x and img. At a maximum or minimum point he noted that dv = 0, so that the equation defining the curve had a double root (img and img) at that point. He noted that the two cases could be distinguished by the concavity of the curve, defining the curve to be concave if the difference of the increments ddv (which we would now write as img) was positive, so that the increments dv themselves increased with increasing img. He defined a point where the increments changed from decreasing to increasing to be a point of reversed bending (punctum flexus contrarii, what we now call an inflection point), and remarked that at such a point (if it was a point where dv = 0 also), the equation had a triple root. What he said is easily translated into the language of today, by looking at the equation 0 = f(x + h) − f(x). Obviously, h = 0 is a root. At a maximum or minimum, it is a double root. If the point x yields df = 0 (that is, f′ (x) = 0) but is not a maximum or minimum, then h = 0 is a triple root (f″ (x) = 0).

In 1686, Leibniz was the first to use the phrase osculating circle. He explained the matter thus:

In the infinitely small parts of a curve it is possible to consider not only the direction or inclination or declination, as has been done up to now, but also the change in direction or curvature (flexura), and as the measures of the direction of curves are the simplest lines of geometry having the same direction at the same point, that is, the tangent lines, likewise the measure of curvature is the simplest curve having at the same point not only the same direction but also the same curvature, that is a circle not only tangent to the given curve but, what is more, osculating.3

Leibniz recognized the problem of finding the evolute as that of constructing “not merely an arbitrary tangent to a single curve at an arbitrary point, but a unique common tangent4 of infinitely many curves belonging to the same order.” That meant differentiating with respect to the parameter and eliminating it between the equation of the family and the differentiated equation. In short, Leibniz was the first to discuss what is now called the envelope of a family of curves defined by an equation containing a parameter, that is, a curve tangent to every curve in the family that it intersects.

39.2 The Eighteenth Century: Surfaces

Compared to calculus, differential equations, and analysis in general, differential geometry was not the subject of a large number of papers in the eighteenth century. Nevertheless, there were important advances.

39.2.1 Euler

According to Coolidge (1940, p. 325), Euler's most important contribution to differential geometry came in a 1760 paper on the curvature of surfaces. In that paper he observed that different planes cutting a surface at a point would generally intersect it in curves having different curvatures, but that the two planes for which this curvature was maximal or minimal would be at right angles to each other. For any other plane, making angle α with one of these planes, the radius of curvature would be

equation

where f and g are the minimum and maximum radii of curvature at the point. Nowadays, because of an 1813 treatise of Pierre Dupin (1784–1873), this formula is written in terms of the curvature 1/r as

equation

where α is the angle between the given cutting plane and the plane in which the curvature is minimal (1/g). The equation obviously implies that in a plane perpendicular to the given plane the curvature would be the same expression with the cosine and sine reversed, or, what is the same, with f and g reversed. Gauss used the formula in Dupin's form, writing it as a formula for the curvature:

equation

where T and V are the maximal and minimal radii of curvature.

Another innovation due to Euler was now-familiar idea of a parameterized surface, in a 1770 paper on surfaces that can be mapped into a plane. The canvas on which an artist paints and the paper on which an engineer or architect draws plans are not only two-dimensional but also flat, having curvature zero. Parameters allow the mathematician or engineer to represent information about any curved surface in the form of functions (t, u) img (x(t, u), y(t, u), z(t, u)). Quantities such as curvature and area are then expressed as functions of the parameters (t, u).

39.2.2 Lagrange

Another study of surfaces, actually a paper in the calculus of variations, was Lagrange's 1762 work on extremal values of integrals.5 The connection with differential geometry is in the problem of minimal surfaces and isoperimetric problems, although he began with the brachistochrone problem (finding the curve of most rapid descent for a falling body). Lagrange found a necessary condition for a surface z = f(x, y) to be the minimal surface having a prescribed boundary.

39.3 Space Curves: The French Geometers

After these “preliminaries” we finally arrive at the traditional beginning of differential geometry, a 1771 paper of Gaspard Monge on curves in space and his 1780 paper on curved surfaces. Monge elaborated Leibniz’ idea for finding the envelope6 of a family of lines, considering a family of planes parameterized by their intersections with the z-axis, and obtained the equation of the surface that is the envelope of the family of planes and can be locally mapped into a plane without stretching or shrinking.

39.4 Gauss: Geodesics and Developable Surfaces

With the nineteenth century, differential geometry entered on a period of growth and has continued to reach new heights for two full centuries. The first mathematician to be mentioned is Gauss, who during the 1820s was involved in mapping the region of Hannover in Lower Saxony, where Göttingen is located. This mapping had been ordered by King George IV of England, who was also Elector of Hannover by inheritance from his great grandfather George I. Gauss had been interested in geodesy for many years (Reich, 1977, pp. 29–34) and had written a paper in response to a problem posed by the Danish Academy of Sciences. This paper, which was published in 1825, discussed conformal mapping, that is, mappings that are a pure magnification at each point, so that angles are preserved and the limiting ratio of the actual distance between two points to the map distance between them as one of them approaches the other is the same for approach from any direction.

Involvement with the mapping project inspired Gauss to reflect on the mathematical aspects of developing a curved surface on a flat page and eventually, the more general problem of developing one curved surface on another—that is, mapping the surfaces so that the ratio that the distance from a given point P to a nearby point Q has to the distance between their images P′ and Q′ tends to 1 as Q tends to P. Gauss apparently planned a full-scale treatise on geodesy but never completed it. Two versions of his major work Disquisitiones generales circa superficies curvas (General Investigations of Curved Surfaces) were written in the years 1825 and 1827. In the preface to the latter Gauss explained the problem he had set: “to find all representations of a given surface upon another in which the smallest elements remain unchanged.” He admitted that some of what he was doing needed to be made more precise through a more careful statement of hypotheses, but wished to show certain results of fundamental importance in the general problem of mapping. He mentioned three ways of defining a surface: first, as the zero set of a function W(x, y, z) of three variables, second as a parameterized mapping (p, q) img (x(p, q), y(p, q), z(p, q)), and third as the graph of a function z = f(x, y). The third case, he pointed out, was merely a specialization of either of the first two.

To determine the extent to which a surface curves, Gauss represented any line in space by a point on a fixed sphere of unit radius: the endpoint of the radius parallel to the line.7 This idea, he said, was inspired by the use of the celestial sphere in geometric astronomy. When the line is the normal line through a point of the surface, the result is a mapping from the surface to the unit sphere, so that the sphere and the surface have parallel normal lines at corresponding points. Obviously a plane maps to a single point under this procedure, since all of its normal lines are parallel to one another. Gauss proposed to use the area of the portion of the sphere covered by this map as a measure of curvature of the surface in question. He called this area the total curvature of the surface. He then attached a sign to this total curvature by specifying that it was to be positive if the surface was convex in both of two mutually perpendicular directions and negative if it was convex in one direction and concave in the other (like a saddle). Gauss gave an informal discussion of this question in terms of the side of the surface on which an oriented normal line was pointing. When the quality of convexity varied in different parts of a surface, Gauss said, a still more refined definition was necessary, which he found it necessary to omit. Along with the total curvature he defined what we would call its density function and he called the measure of curvature, namely the ratio of the area of a local neighborhood on the sphere to the area of the local neighborhood on the surface corresponding to the same parameter values under the two mappings. He denoted this measure of curvature k, a positive or negative sign being attached in accordance with the principles mentioned above. The simplest example is provided by a sphere of radius R, any region of which projects to the similar region on the unit sphere. The ratio of the areas is k = 1/R2, which is therefore the measure of curvature of a sphere at every point.

Thus, in discussing curvature when the surface is given by parameters, Gauss used two mappings from the parameter space (p, q) into three-dimensional space. The first was the mapping onto the surface itself:

equation

The second was the mapping

equation

to the unit sphere, which takes (p, q) to the three direction cosines of the normal to the surface at the point (x(p, q), y(p, q), z(p, q)).

From these preliminaries, Gauss was able to derive very simply what he himself described as “almost everything that the illustrious Euler was the first to prove about the curvature of curved surfaces.” In particular, he showed that his measure of curvature k was the reciprocal of the product of the two principal radii of curvature that Euler called f and g. He then went on to consider more general parameterized surfaces. Here he introduced the now-standard quantities E, F, and G, given by

equation

and what is now called the first fundamental form for the square of the element of arc length:

equation

It is easy to compute that the element of area—the area of an infinitesimal parallelogram whose sides are img and img—is just Δ dp dq, where img. Gauss denoted the analogous expression for the mapping into the unit sphere ((p, q) img (X(p, q), Y(p, q), Z(p, q))), by

(39.1) equation

This quadratic form—or a multiple of it, since definitions vary—is called the second fundamental form. As just described, it is produced using the mapping into the unit sphere to generate the element of surface area on that sphere in terms of the parameters. This parameterization can be used to generate an oriented normal line, which must be parallel to the line from the origin to the image point on the unit sphere where the normal is calculated. If that normal points outward from the unit sphere, the curvature of the surface at the corresponding point is positive. If it points inward, that curvature is negative.

The element of area on the unit sphere is img. Hence, up to the choice of sign, the measure of curvature—what is now called the Gaussian curvature and denoted k—is

equation

In a very prescient remark that was later to be developed by Riemann, Gauss noted that “for finding the measure of curvature, there is no need of finite formulæ, which express the coordinates x, y, z as functions of the indeterminates p, q; but that the general expression for the magnitude of any linear element is sufficient.” The idea is that the geometry of a surface is to be built up from the infinitesimal level using the parameters, not derived from the metric imposed on it by its position in Euclidean space. That is the essential idea of what is now called a differentiable manifold.

Summary of the History up to this Point. From the work of Euler, it was clear that a critical point of a function z = f(x, y) (a point (x, y) where the two partial derivatives are zero) is an extremum if the two principal radii of curvature at the point have the same sign and a saddle point if they have opposite sign. Gauss gave an interpretation to the absolute value of the product of these two curvatures as the limit of the ratio of the area of a small patch of surface near the point to the area of the trace of its unit normal over that patch. Thus geometry and analysis came together very fruitfully in this work. The connection between area and curvature was to have profound consequences.

39.4.1 Further Work by Gauss

It is also clear from Gauss' correspondence (Klein, 1926, p. 16) that Gauss already realized that non-Euclidean geometry was consistent. In fact, the question of consistency did not trouble him; he was more interested in measuring large triangles to see if the sum of their angles could be demonstrably less than two right angles. If so, what we now call hyperbolic geometry might be more convenient for physics than Euclidean geometry.

Gauss considered the possibility of developing one surface on another, that is, mapping it in such a way that lengths are preserved on the infinitesimal level. If the mapping is img, then by composition, u, img, and imgare all functions of the same parameters that determine x, y, and z, and they generate functions E′, F′, and G′ for the second surface that must be equal to E, F, and G at the corresponding points, since that is what is meant by developing one surface on another. But since he had just derived an expression for the measure of curvature that depended only on E, F, G and their partial derivatives, he was able to state the profound result that has come to be called his theorema egregium (outstanding theorem): If a curved surface is developed on any other surface, the measure of curvature at each point remains unchanged. This theorem implies that surfaces that can be developed on a plane, such as a cone or cylinder, must have Gaussian curvature 0 at each point.

With the first fundamental form Gauss was able to derive a pair of differential equations that must be satisfied by geodesic lines, which he called shortest lines,8 and prove that a geodesic circle—the set of endpoints of geodesics originating at a given point and having a given length—intersects each geodesic at a right angle. This result was the foundation for a generalized theory of polar coordinates on a surface, using p as the distance along a geodesic from a variable point to a pole of reference and q as the angle between that geodesic and a fixed geodesic through the pole. This topic very naturally led to the subject of geodesic triangles, formed by joining three points to one another along geodesics. Since he had shown earlier that the element of surface area was

equation

and that this expression was particularly simple when one of the sets of coordinate lines consisted of geodesics (as in the case of a sphere, where the lines of longitude are geodesics), the total curvature of such a triangle was easily found for a geodesic triangle and turned out to be

equation

where A, B, and C are the angles of the triangle, expressed in radians. For a plane triangle this expression is zero. For a spherical triangle it is, not surprisingly, the area of the triangle divided by the square of the radius of the sphere. In this way, area, curvature, and the sum of the angles of a triangle were shown to be linked on curved surfaces. This result was the earliest theorem on global differential geometry, since it applies to any surface that can be triangulated. In its modern version, it relates curvature to the topological property of the surface as a whole known as the Euler characteristic mentioned in the previous chapter. It is called the Gauss–Bonnet theorem after Pierre Ossian Bonnet (1819–1892), who introduced the notion of the geodesic curvature of a curve on a surface (that is, the tangential component of the acceleration of a point moving along the curve with unit speed)9 and generalized the formula to include this concept.

39.5 The French and British Geometers

In France, differential geometry was of interest for a number of reasons connected with physics. In particular, it seemed applicable to the problem of heat conduction, the theory of which had been pioneered by such outstanding mathematicians as Jean-Baptiste Joseph Fourier (1768–1830), Siméon-Denis Poisson (1781–1840), and Gabriel Lamé (1795–1870), since isothermal surfaces and curves in a body were a topic of primary interest. It also applied to the theory of elasticity, studied by Lamé and Sophie Germain (1776–1831), among others. Lamé developed a theory of elastic waves that he hoped would explain light propagation in an elastic medium called ether. Sophie Germain noted that the average of the two principal curvatures derived by Euler would be the same for any two mutually perpendicular planes cutting a surface. She therefore recommended this average curvature as the best measure of curvature. Her idea is useful in elasticity theory,10 but turns out not to be so useful for pure geometry.11 Joseph Liouville (1809–1882) proved that conformal maps of three-dimensional regions are far less varied than those in two dimensions, being necessarily either inversions or similarities or rigid motions. He published this result in the fifth edition of Monge's book on the applications of analysis to geometry. In contrast, a mapping img is conformal if and only if one of the functions u(x, y) ± iv(x, y) is analytic. As a consequence, there is a rich supply of conformal mappings of the plane.

After Newton, differential geometry languished in Britain until the nineteenth century, when William Rowan Hamilton (1805–1865) published papers on systems of rays, building the foundation for the application of differential geometry to differential equations. Another British mathematician, George Salmon (1819–1904), made the entire subject more accessible with his famous textbooks Higher Plane Curves (1852) and Analytic Geometry of Three Dimensions (1862).

39.6 Grassmann and Riemann: Manifolds

Once the idea of using parameters to describe a surface has been grasped, the development of geometry can proceed algebraically, without reference to what is possible in three-dimensional Euclidean space. This idea was developed in the mid-nineteenth century by a number of German and Italian mathematicians.

39.6.1 Grassmann

One mathematician who took the algebraic point of view in geometry was Hermann Grassmann (1809–1877), a secondary-school teacher, who wrote a philosophically oriented mathematical work published in 1844 under the title Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Lineal Extensions, a New Branch of Mathematics). This work, which developed ideas Grassmann had conceived earlier in a work on the ebb and flow of tides, contained much of what is now regarded as multilinear algebra. What we call the coefficients in a linear combination of vectors Grassmann called the numbers by means of which the quantity was derived from the other quantities. He introduced what we now call the tensor product and the wedge product for what he called extensive quantities. He referred to the tensor product simply as the product and the wedge product as the combinatory product. The tensor product of two extensive quantities ∑αrer and ∑βses was

equation

The combinatory product was obtained by applying to this product the rule that [er, es] = − [es, er] (antisymmetrizing). The determinant is a special case of the combinatory product. Grassmann remarked that when the factors are “numerically related” (which we call linearly dependent), the combinatory product would be zero. When the basic units er and es were entirely distinct, Grassmann called the combinatory product the outer product to distinguish it from the inner product, which is still called by that name today and amounts to the ordinary dot product when applied to vectors in physics. Grassmann remarked that parentheses have no effect on the outer product—in our terms, it is an associative operation.12

Working with these concepts, Grassmann defined the numerical value of an extended quantity as the positive square root of its inner square, exactly what we now call the absolute value of a vector in n-dimensional space. He proved that “the quantities of an orthogonal system are not related numerically,” that is, an orthogonal set of nonzero vectors is linearly independent.

39.6.2 Riemann

Historians of mathematics seem to agree that, because of its philosophical tone and unusual nomenclature, Ausdehnungslehre did not attract a great deal of notice until Grassmann revised it and published a more systematic exposition in 1862. If that verdict is correct, there is a small coincidence in Riemann's use of the term “extended,” which appears to mimic Grassmann's use of the word, and in his focus on a general number of dimensions in his inaugural lecture at the University of Göttingen.

Riemann's most authoritative biographer Laugwitz (1999, p. 223) says that Grassmann's work would have been of little use to Riemann, since for him linear algebra was a trivial subject.13 The inaugural lecture was read in 1854, with the aged Gauss in the audience.14 Although Riemann's lecture “Über die Hypothesen die der Geometrie zu Grunde liegen” (“On the hypotheses that form the basis of geometry”) occupies only 14 printed pages and contains almost no mathematical symbolism—it was aimed at a largely nonmathematical audience—it set forth ideas that had profound consequences for the future of both mathematics and physics. As Hermann Weyl said:

The same step was taken here that was taken by Faraday and Maxwell in physics, the theory of electricity in particular,. . .by passing from the theory of action at a distance to the theory of local action: the principle of understanding the world from its behavior on the infinitesimal level. [Narasimhan, 1990, p. 740]

In the first section, Riemann began by developing the concept of an n-fold extended quantity, asking the indulgence of his audience for delving into philosophy, where he had limited experience. He cited only some philosophical work of Gauss and of Johann Friedrich Herbart, who was mentioned in the previous chapter, Riemann began with the concept of quantity in general, which arises when some general concept can be defined (measured or counted) in different ways. Then, according as there is or is not a continuous transformation from one of the ways into another, the various determinations of it form a continuous or discrete manifold. He noted that discrete manifolds (sets of things that can be counted, as we would say) are very common in everyday life, but continuous manifolds are rare, the spatial location of objects of sense and colors being almost the only examples.

The main part of the lecture was the second part, in which Riemann investigated the kinds of metric relations that could exist in a manifold if the length of a curve was to be independent of its position. Assuming that the point was located by a set of n coordinates x1,. . ., xn (almost the only mathematical symbols that appear in the paper), he considered the kinds of properties needed to define an infinitesimal element of arc length ds along a curve. The simplest function that met this requirement was

equation

where the coefficients aij were continuous functions of position and the expression under the square root is always nonnegative. The next simplest case, which he chose not to develop, occurred when the Maclaurin series began with fourth-degree terms. As Riemann said,

The investigation of this more general type, to be sure, would not require any essentially different principles, but it would be rather time-consuming and cast relatively little new light on the theory of space; and moreover the results could not be expressed geometrically.

For the case in which coordinates could be chosen so that aii = 1 and aij = 0 when ij, Riemann called the manifold flat.

Having listed the kinds of properties space was assumed to have, Riemann asked to what extent these properties could be verified by experiment, especially in the case of continuous manifolds. What he said at this point has become famous. He made a distinction between the infinite and the unbounded, pointing out that while space is always assumed to be unbounded (that is, to have no border), it might very well not be infinite. Then, as he said, assuming that solid bodies exist independently of their position, it followed that the curvature of space would have to be constant, and all astronomical observation confirmed that it could only be zero. But, if the volume occupied by a body varied as the body moved, no conclusion about the infinitesimal nature of space could be drawn from observations of the metric relations that hold on the finite level. “It is therefore quite conceivable that the metric relations of space are not in agreement with the assumptions of geometry, and one must indeed assume this if phenomena can be explained more simply thereby.” Such reasoning plays a role in the theory of relativity, where the rigid body of classical mechanics does not exist.

Riemann evidently intended to follow up on these ideas, but his mind produced ideas much faster than his frail body would allow him to develop them. He died before his 40th birthday with this project one of many left unfinished. He did, however, send an essay to the Paris Academy in response to a prize question proposed (and later withdrawn): Determine the thermal state of a body necessary in order for a system of initially isothermal lines to remain isothermal at all times, so that its thermal state can be expressed as a function of time and two other variables. Riemann's essay was not awarded the prize because its results were not developed with sufficient rigor. It was not published during his lifetime.15

39.7 Differential Geometry and Physics

The work of Grassmann and Riemann was to have a powerful impact on the development of both geometry and physics. One has only to read Einstein's accounts of the development of general relativity to understand the extent to which he was imbued with Riemann's outlook. The idea of geometrizing physics seems an attractive one. The Aristotelian idea of force, which had continued to serve through Newton's time, began to be replaced by subtler ideas developed by the Continental mathematical physicists of the nineteenth century, with the introduction of such principles as conservation of energy and least action. In his 1736 treatise on mechanics, Euler had shown that a particle constrained to move along a surface by forces normal to the surface, but on which no forces tangential to the surface act, would move along a shortest curve on the surface. And when he discovered the variational principles that enabled him to solve the isoperimetric problem, he applied them to the theory of elasticity and vibrating membranes. As he said,

Since the material of the universe is the most perfect and proceeds from a supremely wise Creator, nothing at all is found in the world that does not illustrate some maximal or minimal principle. For that reason, there is absolutely no doubt that everything in the universe, being the result of an ultimate purpose, is amenable to determination with equal success from these efficient causes using the method of maxima and minima. [Euler, 1744, p. 245]

Riemann was searching for connections among light, electricity, magnetism, and gravitation at this time.16 In 1846, Gauss' collaborator Wilhelm Weber (1804–1891) had incorporated the velocity of light in a formula for the force between two moving charged particles. According to Hermann Weyl (Narasimhan, 1990, p. 741), Riemann did not make any connection between that search and the content of his inaugural lecture. Laugwitz (1999, p. 222), however, cites letters from Riemann to his brother which show that he did make precisely that connection. Whatever the case, four years later Riemann sent a paper17 to the Royal Society in Göttingen in which he made the following remarkable statement:

I venture to communicate to the Royal Society a remark that brings the theory of electricity and magnetism into a close connection with the theory of light and heat radiation. I have found that the electrodynamic effects of galvanic currents can be understood by assuming that the effect of one quantity of electricity on others is not instantaneous but propagates to them with a velocity that is constant (equal to that of light within observational error).

39.8 The Italian Geometers

The political unification of Italy in the mid-nineteenth century was accompanied by a surge of mathematical activity even greater than the sixteenth-century work in algebra. Gauss had analyzed a general surface by using two parameters and introducing six functions: the coefficients of the first and second fundamental forms. The question naturally arises whether a surface can be synthesized from any six functions regarded as the coefficients of these forms. Do they determine the surface, up to the usual Euclidean motions of translation, rotation, and reflection that can be used to move a set of axes to a prescribed position and orientation? Such a theorem does hold for curves, as was established by two French mathematicians, Jean Frenet (1816–1900) and Joseph Serret (1818–1885), who gave a set of equations—the Frenet–Serret18 equations—determining the curvature and torsion19 of a curve in three-dimensional spaces. A curve can be reconstructed from its curvature and torsion up to translation, rotation, and reflection. A natural related question is: Which sets of six functions, regarded as the components of the two fundamental forms, can be used to construct a surface? After all, one needs generally only three functions of two parameters to determine a surface, so that the six given by Gauss cannot be independent of one another.

In an 1856 paper, Gaspare Mainardi (1800–1879) provided consistency conditions in the form of four differential equations, now known as the Mainardi–Codazzi equations,20 which must be satisfied by the six functions E, F, G, D, D′, and D” if they are to be the components of the first and second fundamental forms introduced by Gauss. Mainardi had learned of Gauss' work through a French translation, which had appeared in 1852. These same equations were discovered by Delfino Codazzi (1824–1875) two years later, using an entirely different approach, and helped him to win a prize from the Paris Academy of Sciences. Codazzi published these equations only in 1883.

When Riemann's lecture was published in 1867, the year after his death, it became the point of departure for a great deal of research in Italy.21 One who worked to develop these ideas was Riemann's friend Enrico Betti, who tried to get Riemann a chair of mathematics in Palermo. These ideas led Betti to the notion of the connectivity of a surface. On the simplest surfaces, such as a sphere, every closed curve is the boundary of a region. On a torus, however, the circles of latitude and longitude are not boundaries. These ideas belong properly to topology, and were discussed in the preceding chapter. In his fundamental work on this subject, Henri Poincaré named the maximum number of independent nonboundary cycles in a surface the Betti number of the surface, a concept that is now generalized to n dimensions. The nth Betti number is the rank of the nth homology group.

Another Italian mathematician who extended Riemann's ideas was Eugenio Beltrami (1835–1900), whose 1868 paper on spaces of constant curvature contained a model of a three-dimensional space of constant negative curvature. Beltrami had previously given the model of a pseudosphere to represent the hyperbolic plane, which will be discussed in the next chapter. It was not obvious before his work that three-dimensional hyperbolic geometry and a three-dimensional manifold of constant negative curvature were basically the same thing. Beltrami also worked out the appropriate n-dimensional analogue of the Laplacian img, which plays a fundamental role in mathematical physics. By working with an integral considered earlier by Jacobi (see Klein, 1926, Vol. 2, p. 190), Beltrami arrived at the operator

equation

where, with the notation slightly modernized, the Riemannian metric is given by the usual img, and a denotes the determinant det (aij). The generalized operator is now referred to as the Laplace–Beltrami operator on a Riemannian manifold.

39.8.1 Ricci's Absolute Differential Calculus

The algebra of Grassmann and its connection with Riemann's general metric on an n-dimensional manifold was not fully codified until 1901, in “Méthodes de calcul différentiel absolu et leurs applications” (“Methods of absolute differential calculus and their applications”), published in Mathematische Annalen in 1901, written by Gregorio Ricci-Curbastro (1853–1925) and Tullio Levi-Civita (1873–1941). This article contained the critical ideas of tensor analysis as it is now taught. The absoluteness of the calculus consisted in the great generality of the transformations that it permitted, showing how differential forms changed when coordinates were changed. Although Ricci-Curbastro competed in a prize contest sponsored that year by the Accademia dei Lincei, he was not successful. Some of the judges regarded his absolute differential calculus as superfluous to the end it was designed for.22

The following year, Luigi Bianchi (1873–1928) published “Sui simboli a quattro indice e sulla curvatura di Riemann” (“On quadruply-indexed symbols and Riemannian curvature”), in which he gave the relations among the covariant derivatives of the Riemann curvature tensor, which he derived by a direct method for manifolds of constant curvature, not following the route of Ricci-Curbastro and Levi-Civita. The Bianchi identity was later to play a crucial role in general relativity, assuring local conservation of energy when Einstein's gravitational equation is assumed.

Problems and Questions

Mathematical Problems

39.1 Find the radius of curvature of the parabola y = x2 at the point (1, 1). (The center of curvature lies along the normal line at that point, which has equation x + 2y = 3.)

39.2 Find the first fundamental form of the hyperbolic paraboloid z = (x2y2)/a at each point using x and y as parameters.

39.3 Find the Gaussian curvature of the hyperbolic paraboloid described in the previous problem.

Historical Questions

39.4 What motives did Huygens and Newton have for studying involutes, evolutes, and curvature?

39.5 What significant increase in the algebraization of geometry was promoted by Euler?

39.6 In what ways did Riemann anticipate much of modern physics?

Questions for Reflection

39.7 We saw earlier (Chapter 27) that Omar Khayyam criticized ibn al-Haytham for discussing a line that moves while remaining perpendicular to another line. Yet Newton used exactly the same language when discussing the center of curvature, and no one seems to have objected. What difference in mathematical cultures does this contrast in points of view signify?

39.8 Riemann raised the issue whether space was infinite or finite. In 1976, the mathematician John Milnor (b. 1931), in an address to the Northeastern Section of the Mathematical Association of America, asked whether space is simply connected. One way of investigating this question would be to look for doubly periodic star patterns, such as one would see if the celestial sphere were actually a torus. In the discussion that followed, a member of the audience asked what reason we have for thinking that space is even orientable. Might it not resemble a projective plane? How could one investigate this question empirically?

39.9 How much of modern differential geometry could be cast in the language of Euclidean geometry? How would you describe, for example, the hyperbolic paraboloid whose equation is az = x2y2?

Notes

1. Despite the legend that Galileo observed a chandelier swinging and noticed that all its swings, whether wide or short, required the same amount of time to complete, that observation holds true for circular arcs only approximately and only for small amplitudes, as anyone who has done the experiment in high-school physics will have learned.

2. The master's thesis of Robert W. Katsma at California State University at Sacramento in the year 2000 was entitled “An analysis of the failure of Huygens' cycloidal pendulum and the design and testing of a new cycloidal pendulum.” Katsma was granted patent 1992-08-18 in Walla Walla County for a cycloidal pendulum. However, the theoretical consensus is that such devices only decrease the accuracy of a good clock.

3. Literally, kissing.

4. The tangent was not necessarily to be a straight line.

5. Œuvres de Lagrange, T. 1, pp. 335–362.

6. The envelope of a family of surfaces is a surface that is tangent to each of them.

7. An oriented line is meant here, since there are obviously two opposite radii parallel to the line. Gauss surely knew that the order of the parameters could be used to fix this orientation.

8. According to Klein (1926, Vol. 2, p. 148), the term geodesic was first used by Joseph Liouville (1809–1882) in 1850. Klein cites an 1893 history of the term by Paul Stäckel (1862–1919) as source.

9. According to Struik (1933, 20, pp. 163, 165), even this concept was anticipated by Gauss in an unpublished paper of 1825 and followed up on by Ferdinand Minding (1806–1885) in a paper in the Journal für die reine und angewandte Mathematik in 1830.

10. In particular, her concept of the average curvature plays a role in the Navier–Stokes equations.

11. The average curvature must be zero on a minimal surface, however.

12. To avoid confusing the reader who knows that the cross product is not an associative product, we note that the outer product applies only when each of the factors is orthogonal to the others. In three dimensional space the cross product of three such vectors, however they are grouped, is always zero. The wedge product, however, is associative.

13. One can't help wondering about the multilinear algebra that Grassmann was developing. The recognition of this theory as an essential part of geometry is explicit in Felix Klein's 1908 work on elementary geometry from a higher viewpoint, but Riemann apparently did not make the connection.

14. At the time of the lecture, Gauss had less than a year of life remaining. Yet his mind was still active, and he was very favorably impressed by Riemann's performance.

15. Klein (1926, Vol. 2, p. 165) notes that very valuable results were often submitted for prizes at that time, since professors were so poorly paid.

16. His lecture was given nearly a decade before Maxwell discovered his famous equations connecting the speed of light with the propagation of electromagnetic waves.

17. This paper was later withdrawn, but was published after his death (Narasimhan, 1990, pp. 288–293).

18. Frenet gave six equations for the direction cosines of the tangent and principal normal to the curve and its radius of curvature. Serret gave the full set of nine now called by this name, which are more symmetric but contain no more information than the six of Frenet.

19. The torsion of a curve measures its tendency to move out of the plane of its tangential and principal normal vectors.

20. The Latvian mathematician Karl Mikhailovich Peterson (1828–1881) published an equivalent set of equations in Moscow in 1853, but they went unnoticed for a full century.

21. Riemann went to Italy for his health and died of tuberculosis in Selasca. He was in close contact with Italian mathematicians and even published a paper in Italian.

22. The same sort of criticism was leveled by Weierstrass against the work of Hamilton in quaternions.