George Friedrich Bernhard Riemann founded Riemannian geometry, which is of fundamental importance to both mathematics and physics.

Mathematics–Riemannian Geometry and the Riemann Hypothesis

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Riemann studied under Carl Gauss at Göttingen, a university in Germany and became professor there. Riemann created Riemannian geometry, which is a form of differential non-Euclidean geometry where space is everywhere positively curved. In Riemann’s geometry, parallel lines do not exist, the angles of a triangle do not add up to 180°, and perpendiculars to the same line converge. It provided Einstein with a basis for his general theory of relativity. 

The Riemann hypothesis, about the complex numbers that are roots of a certain transcendental equation, remains one of the unsolved problems of mathematics(1) and is one of the seven millennium problems which has a $1 million prize for its solution. In the opinion of many mathematicians, the Riemann hypothesis, and its extension to general classes of L-functions, is probably the most important open problem in pure mathematics today.(2) (see end of conclusion for this section for more information about each of the millennium problems).

Non-Euclidean Geometry

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“The creation of non-Euclidean geometry was the most consequential and revolutionary step in mathematics since Greek times . . . (because) the axioms of geometry are our basic facts about physical space, and vast branches of mathematics and physical science use the properties of Euclidean geometry.” pgs. 879-880 in Morris Kline’s Mathematical Thought from Ancient to Modern Times.

As presented in the biographies, four people are usually recognized for the discovery of non-Euclidean geometries – Gauss (1777-1855), Lobachevsky (1792-1856), Bolyai (1802-1860), and Riemann (1826-1866).

Gauss invented the term “Non-Euclidean Geometry” but never published anything on the subject.

Gauss’s work in geometry led to the discovery of the two different kinds of non-Euclidean geometry: hyperbolic geometry, which is associated with the Russian mathematician Lobachevsky and the Hungarian mathematician Bolyai, and elliptical geometry, which was discovered by Riemann, a student of Gauss. Although Gauss himself did not develop the complete mathematics of non-Euclidean geometry, he constructed the mathematical framework around which the main body of non-Euclidean geometry had to be built.(3)

To understand the need for non-Euclidean geometry, you must look at the weakest link in the chain of postulates on which Euclidean geometry is built—the famous parallel postulate (the fifth postulate), which states the following:

Given a straight line and a point outside this line, one and only one straight line (in the plane determined by the given line and point) can be drawn through the point parallel to the given line.

As far back as the fifteenth century, mathematicians were uncomfortable with this postulate and so they tried to prove it using the other postulates without success. Gauss was thus attracted to it as a challenge, but he soon became convinced that the parallel postulate is, indeed, a postulate and that it, therefore, cannot be proved; it must be accepted as a truth if one works with Euclidean geometry.(4)

Gauss, like Newton, abhorred controversy and refused to publicize mathematical discoveries that might lead to a conflict of ideas between him and other mathematicians. His letters and his diaries show that he had made many mathematical discoveries that he did not publish.(5) In 1800, for example, he discovered certain important functions known as “elliptic functions” and by 1816 he had extended his early geometry to encompass most of the features of the non-Euclidean geometries discovered later by Lobachevsky, Bolyai, and Riemann.(6) Gauss’s diaries show that he did not accept the Kantian thesis that the Euclidean geometry of space must be accepted as an absolute, a priori truth.(7) He insisted that the nature of the correct geometry of space must be determined by observation as an experimental fact. In line with this approach, Gauss proposed an experiment which could not be performed with the crude technology of that era. He suggested that the three angles of a triangle, formed by the three lines connecting three different stars, be measured to see if their sum is 180 degrees as demanded by Euclidean geometry. This experiment was never performed, but it underscores Gauss’s rejection of an a priori concept about the correct geometry of space.(8)

This discussion leads to the other great mid-nineteenth-century geometers Lobachevsky, Bolyai, and Riemann, who continued Gauss’s work on non-Euclidean geometry and brought it to its modern form. The works of Lobachevsky and Bolyai did not stem directly from Gauss’s work but rather from the rejection of Euclid’s fifth postulate (the parallel postulate), which had troubled mathematicians from the time of its pronouncement by Euclid to the discovery of non-Euclidean geometry in the nineteenth century. For almost two thousand years, outstanding mathematicians had tried to prove it as a theorem rather than accepting it as an independent postulate so that the notions of a non-Euclidean geometry were implicit in the failure of these proofs.(9) In his A Concise History of Mathematics, Dirk Struick notes that attempts to derive Euclid’s fifth postulate go back to Ptolemy, such mathematicians as Nasir-al-din in the Middle Ages and Lambert and Legendre in the eighteenth and nineteenth centuries, respectively, devoted much time to it.(10)

These attempts were not abandoned until after Gauss had developed his surface geometry which convinced Gauss that non-Euclidean geometries are valid but can be constructed only if the fifth postulate of Euclid is replaced by another parallel postulate. Why Gauss stopped at this point, going no further than introducing the label “non-Euclidean geometry” is not clear; he certainly had the mathematical ability to do so.

If Gauss was deterred from developing a complete theory of non-Euclidean geometry because he doubted its reality or its practical usefulness, Lobachevsky, Bolyai, and Riemann were quite the opposite. Being unsatisfied with Euclid’s parallel postulate, each, in his own way, decided to see how far he could go without Euclid’s fifth postulate. Simple geometrical considerations convinced each one that leaving the postulate out altogether would not do because it would be impossible to prove certain basic geometrical theorems.(11) The parallel postulate makes it possible for one to prove, for example, that the sum of the three angles of a triangle is 180 degrees and that the circumference of a circle equals 2 times pi ( about 3.14) times its radius r. But the restrictions imposed on Euclidean geometry by its basic elements, the line and the point, make it an ideal intellectual concept rather than a real one because neither a point nor a line had any reality.(12) Thus Euclidean geometry ultimately would have to give way to a real geometry which could only be produced by replacing Euclid’s fifth postulate by another “parallel” postulate. Bolyai and Lobachevsky, independently, carried this out in a way which led to what we now call hyperbolic geometry. Riemann, by contrast, replaced Euclid’s fifth postulate in quite a different way which led to what we now call elliptical geometry.

Both Bolyai and Lobachevsky replaced Euclid’s parallel postulate by the following statement:

Given a straight line and a point outside this line, an infinite number of straight lines can pass through the given point parallel to the given straight line.

This leads to a self-consistent non-Euclidean geometry called Lobachevskian or hyperbolic geometry.

This parallel postulate cannot apply to a flat surface nor to what we ordinarily call straight lines on a plane. The concept of a straight line must be replaced by the concept of the shortest distance between two points (called a geodesic). In Lobachevskian geometry, straight lines are curves. Triangles consist of three such curves with their concave sides out. In a triangle of this sort, the angles are smaller than they would be if the sides were straight in the Euclidean sense. Thus, in Lobachevskian geometry, the sum of the three angels of a triangle is smaller than 180 degrees and the circumference of a circle is larger than 2 times pi (about 3.14) times its radius. A saddle surface, which has two concave sides (no convex or flat side), is an excellent example of a hyperbolic surface. (see illustration below)

Riemann’s contributions to the development of non-Euclidean geometries stemmed, in part, from Gauss’s tutelage at the University of Göttingen, where Riemann earned his Ph.D. degree in mathematics in 1851. Riemann, like Lobachevsky, abandoned Euclid’s parallel postulate, replacing it by the postulate that no parallel lines exist. Indeed, all straight lines intersect in Riemannian (elliptical) geometry. The geometry on the surface of a sphere is an example of elliptical geometry; “straight lines” (shortest distances) on a sphere are arcs of great circles (circles whose radii equal the radius of the sphere) and they all intersect each other. Because the three sides of a triangle on a sphere are the arcs of three great circles, the sum of the three angles of such a triangle is greater than 180 degrees. One can also prove that the circumference of a circle on a sphere is always smaller than 2 times its radius. The circles of longitude (the meridian circles) on the earth’s surface are the equivalent of straight lines and all intersect each other at the north and south poles.

Whereas Euclidean geometry imposes exact arithmetic relations among the geometrical entities that define figures such as triangles (the sum of the angles must equal exactly 180 degrees) and circles (circumferences must equal exactly 2 pi (about 3.14) times their radii), Lobachevskian and Riemannian geometries allow a range of values for the sum of angles of a triangle and for the relationship between a circle’s circumference and its radius.(13) Thus, Euclidean geometry is like a line on a plane that divides the points on the upper half from those on the lower half. Euclidean geometry can be pictured as the limiting case of both Lobachevskian and Riemannian geometries; as one passes from one Riemannian geometry to the next, in infinitesimal steps, in a direction such that in each succeeding example of Riemannian geometry the sum of the three angles of a typical triangle decreases, we instantaneously pass through Euclidean geometry when the sum equals 180 degrees.(14) You then enter Lobachevskian geometry for which the three angle sum is less than 180 degrees and decreases as we move along.

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