September 17, 2020 marked the 194th anniversary of the birth of German mathematician Bernhard Riemann. Why should we care? Well, Riemann was the first person to suggest using dimensions higher than three or four to describe physical reality. And, it was Riemann's work that provided Albert Einstein with the mathematical framework he needed to express the ideas of the Theory of General Relativity.
Who was Bernhard Riemann?
Riemann was the second of six children born to a Lutheran pastor in the village of Breselenz in what was then the Kingdom of Hanover. Intending to follow in his father's footsteps, Riemann studied theology, however, when his father scraped together enough money to send Riemann to the famed University of Göttingen, Riemann began to study mathematics under no less a personage than Carl Friedrich Gauss.
Gauss delighted in his new student, and he described Riemann's Ph.D. thesis on complex variables as the work of someone with "a gloriously fertile originality." With Riemann attempting to gain a faculty position at the university, Gauss encouraged him to reformulate the foundations of geometry which had last been laid down by the Greek mathematician Euclid 2,000 years earlier.
In 1854, Riemann's lecture called, "On the Hypotheses Which Lie at the Foundations of Geometry," founded the field of Riemannian geometry. In the lecture, Riemann introduced the concepts of the metric tensor, curved spaces of arbitrary dimension, geodesics, and the curvature tensor which now bears his name.
Riemannian geometry versus Euclidian geometry
Euclidean geometry concerns flat, 2D spaces. It contains five postulates (axioms):
1. A straight line segment may be drawn from any given point to any other.
2. A straight line may be extended to any finite length.
3. A circle may be described with any given point as its center and any distance as its radius.
4. All right angles are congruent (equal to one another).
5. Through any given point not on a line there passes exactly one line parallel to that line in the same plane.
Riemannian Geometry concerns curved, 3D surfaces such as a cylinder or a sphere, and it threw out Euclid's fifth postulate, also known as the Parallel Postulate, and forever changed the second postulate. In Euclidean geometry, two parallel lines are taken to be everywhere equidistant, while in Riemannian geometry, there are no lines parallel to a given line because, in Riemannian geometry, parallel lines do not exist.
In Riemannian geometry, the shortest curve between any pair of points on a curved surface is called a minimal geodesic. You can find a minimal geodesic between two points on a curved surface by stretching a rubber band between them.
When you do that, you will notice that sometimes there is more than one minimal geodesic between the two points. For example, there are many minimal geodesics between the north and south poles on a globe, and these correspond to longitude lines on Earth. Many parallel lines pass through the two points at the poles, and this contradicts Euclid's Parallel Postulate.
If you draw a triangle or a circle on a curved surface, estimating the length of the hypotenuse of the triangle, or estimating the circumference of the circle and the area inside the circle depends on the amount that the surface is curved.
How much is a surface curved?
There are many different shapes that curved surfaces can take, they can be cylinders, spheres, paraboloids, or tori.
Surfaces can have positive, zero, or negative curvature.
In Euclidean geometry, the sum of the angles of a triangle is 180 degrees. In Riemannian geometry, the sum of the angles of a spherical triangle is the sum of the angles plus the area of the triangle.
Riemannian geometry also concerns the study of higher dimensions of spaces. Near Earth, the universe looks like three-dimensional Euclidean space. However, near very heavy objects such as stars and black holes, space is curved. That means that there are pairs of points in the universe that have more than one minimal geodesic between them.
Gravitational lensing occurs when light from a distant source is bent by the curvature of space, and the amount of bending is one of the predictions of the General Theory of Relativity. The amount that space is curved can be estimated by using Riemannian geometry, and astronomers can then estimate the mass of a star or black hole based on the amount of gravitational lensing. Today, Riemannian geometers are looking for the relationship between the curvature of a space and its actual shape.
Win $1 million
In 2018, 89-year-old mathematician emeritus at The University of Edinburgh Michael Atiyah presented a proof to one of the great unsolved problems in mathematics known as the Riemann Hypothesis. Atiyan's proof was ultimately shown to be incorrect.
It is one of the seven Millennium Prize Problems which were established by the Clay Mathematics Institute in 2000. Anyone solving any one of these problems wins $1 million, and to date, only one Millennium Prize problem has been solved, the Poincare Conjecture which was solved in 2003 by the Russian mathematician Grigori Perelman, who declined the prize money.
Riemann's Hypothesis concerns prime numbers, such as two, three, five, seven, and 11. A prime can only be divided by the number one or itself. As you go up the number line, primes become less and less frequent and are separated by ever-larger gaps.
Riemann proposed that the way to understand the primes' distribution was by analyzing a different set of numbers, the zeroes of a function called the Riemann zeta function. It has both real and imaginary inputs, and if we remember from high school math, imaginary numbers are of the form square root of -1.
Riemann used the zeta function to come up with a formula for calculating how many primes there are up to a certain point, and at what intervals they occur. However, Riemann's formula only holds if the real parts of the zeta function zeroes are all equal to one-half.
For the first few primes, Reimann proved this property, and after the advent of computers during the 1950s, the theory has been computationally shown to work for many prime numbers. However, the theory remains to be formally proved and to be proven out to infinity. Such a proof would have important applications in the field of cryptography.
In 1862 Riemann developed tuberculosis, and he traveled to Italy three times to convalesce. In 1866 on his third trip, Riemann died at only 39-years-of-age. Back in Germany at his office in Göttingen, Riemann's overly efficient housekeeper threw out all his papers, thus possibly disposing of a groundbreaking theory.
Had Riemann lived longer, or if his housekeeper had been less scrupulous, who knows what other wondrous mathematical discoveries he could have made.