Bernhard Riemann

Georg Friedrich Bernhard Riemann (pronounced REE mahn or in ; September 17, 1826 – July 20, 1866) was a German mathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity.

Influence
Riemann was arguably the most influential mathematician of the middle of the nineteenth century. His published works are a small volume only, but opened up research areas combining analysis with geometry.

These would subsequently be major parts of the theories of Riemannian geometry, algebraic geometry and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics was foundational in topology, and in the twenty-first century is still being applied in novel ways to mathematical physics.

Riemann worked in real analysis, where he is also a major figure. Besides defining the Riemann integral, by means of Riemann sums, he developed a theory of trigonometric series that are not Fourier series, a first step in generalized function theory, and studied the Riemann-Liouville differintegral.

He made some of the most famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he introduced the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful heuristic, rather than a rigorous method, and its justification took at least a generation. His work on monodromy and the hypergeometric function in the complex domain made a great impression, and established a basic way of working with functions, by consideration only of their singularities.

Early life
Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is today Germany. His father Friedrich Bernhard Riemann was a poor Lutheran pastor in Breselenz. Friedrich Riemann fought in the Napoleonic Wars. Georg's mother also died before her children were grown. Bernhard was the second of six children. He was a shy boy and suffered from numerous nervous breakdowns. From a very young age, Riemann exhibited his exceptional skills, such as fantastic calculation abilities, but suffered from timidity and had a fear of speaking in public.

Middle life
In high school, Riemann studied the Bible intensively. But his mind often drifted back to mathematics and he even tried to prove mathematically the correctness of the book of Genesis. His teachers were amazed by his genius and by his ability to solve extremely complicated mathematical operations. He often outstripped his instructor's knowledge. In 1840 Bernhard went to Hanover to live with his grandmother and visit the Lyceum. After the death of his grandmother in 1842 he went to the Johanneum in Lüneburg. In 1846, at the age of 19, he started studying philology and theology, in order to become a priest and help with his family's finances.

In 1847 his father, after scraping together enough money to send Riemann to university, allowed him to stop studying theology and start studying mathematics. He was sent to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least squares.

In 1847 he moved to Berlin, where Jacobi, Dirichlet and Steiner were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.

Later life
Riemann held his first lectures in 1854, which not only founded the field of Riemannian geometry but set the stage for Einstein's general relativity. There was an unsuccessful attempt to promote Riemann to extraordinary professor status at the University of Göttingen in 1857, but from that attempt Riemann was finally granted a regular salary. In 1859, following Dirichlet's death he was promoted to head the Mathematics department at Göttingen. He was also the first to propose the theory of higher dimensions, which highly simplified the laws of physics. In 1862 he married Elise Koch and had a daughter. He died of tuberculosis on his third journey to Italy in Selasca (now a hamlet of Ghiffa on Lake Maggiore).

Euclidean geometry versus Riemannian geometry
Gauss asked his student Riemann in 1853 to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions. When he finally delivered his lecture in 1854, the mathematical public received it with enthusiasm.

The subject founded by this work is Riemannian geometry. Riemann had found the correct way to extend into n dimensions the differential geometry of surfaces, for which Gauss himself had proved his theorema egregium. The fundamental object is what is now called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero, the non-zero and constant cases being models of the known non-Euclidean geometries.

Higher dimensions
Riemann's idea was to introduce a collection of numbers at every point in space that would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous metric tensor.

Writings in English

 * 1868. "On the hypotheses which lie at the foundation of geometry" in Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press: 652-61.