**Bernhard Riemann**'s ideas concerning geometry of space had a profound effect on the development of modern theoretical physics. He clarified the notion of integral by defining what we now call the Riemann integral.

- At this time a teacher from a local school named Schulz assisted in Bernhard's education.
- In 1840 Bernhard entered directly into the third class at the Lyceum in Hannover.
- Bernhard seems to have been a good, but not outstanding, pupil who worked hard at the classical subjects such as Hebrew and theology.
- He showed a particular interest in mathematics and the director of the Gymnasium allowed Bernhard to study mathematics texts from his own library.
- On one occasion he lent Bernhard Legendre's book on the theory of numbers and Bernhard read the 900 page book in six days.
- In the spring of 1846 Riemann enrolled at the University of Göttingen.
- This was granted, however, and Riemann then took courses in mathematics from Moritz Stern and Gauss.
- It may be thought that Riemann was in just the right place to study mathematics at Göttingen, but at this time the University of Göttingen was a rather poor place for mathematics.
- Gauss did lecture to Riemann but he was only giving elementary courses and there is no evidence that at this time he recognised Riemann's genius.
- Riemann moved from Göttingen to Berlin University in the spring of 1847 to study under Steiner, Jacobi, Dirichlet and Eisenstein.
- This was an important time for Riemann.
- The main person to influence Riemann at this time, however, was Dirichlet.
- His manner suited Riemann, who adopted it and worked according to Dirichlet's methods.
- Riemann's work always was based on intuitive reasoning which fell a little below the rigour required to make the conclusions watertight.
- It was during his time at the University of Berlin that Riemann worked out his general theory of complex variables that formed the basis of some of his most important work.
- However it was not only Gauss who strongly influenced Riemann at this time.
- Weber had returned to a chair of physics at Göttingen from Leipzig during the time that Riemann was in Berlin, and Riemann was his assistant for 18 months.
- Through Weber and Listing, Riemann gained a strong background in theoretical physics and, from Listing, important ideas in topology which were to influence his ground breaking research.
- Riemann's thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces.
- However, Riemann's thesis is a strikingly original piece of work which examined geometric properties of analytic functions, conformal mappings and the connectivity of surfaces.
- In proving some of the results in his thesis Riemann used a variational principle which he was later to call the Dirichlet Principle since he had learnt it from Dirichlet's lectures in Berlin.
- Riemann's thesis, one of the most remarkable pieces of original work to appear in a doctoral thesis, was examined on 16 December 1851.
- On Gauss's recommendation Riemann was appointed to a post in Göttingen and he worked for his Habilitation, the degree which would allow him to become a lecturer.
- He gave the conditions of a function to have an integral, what we now call the condition of Riemann integrability.
- To complete his Habilitation Riemann had to give a lecture.
- Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry.
- Riemann's lecture Über die Hypothesen welche der Geometrie zu Grunde liegen Ⓣ(On the hypotheses at the foundations of geometry), delivered on 10 June 1854, became a classic of mathematics.
- There were two parts to Riemann's lecture.
- In the first part he posed the problem of how to define an nnn-dimensional space and ended up giving a definition of what today we call a Riemannian space.
- The main point of this part of Riemann's lecture was the definition of the curvature tensor.
- The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in.
- Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented.
- In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data.
- So this brilliant work entitled Riemann to begin to lecture.
- Although only eight students attended the lectures, Riemann was completely happy.
- At this time there was an attempt to get Riemann a personal chair but this failed.
- One of the three was Dedekind who was able to make the beauty of Riemann's lectures available by publishing the material after Riemann's early death.
- The abelian functions paper continued where his doctoral dissertation had left off and developed further the idea of Riemann surfaces and their topological properties.
- He examined multi-valued functions as single valued over a special Riemann surface and solved general inversion problems which had been solved for elliptic integrals by Abel and Jacobi.
- However Riemann was not the only mathematician working on such ideas.
- when Weierstrass submitted a first treatment of general abelian functions to the Berlin Academy in 1857, Riemann's paper on the same theme appeared in Crelle's Journal, Volume 54.
- The Dirichlet Principle which Riemann had used in his doctoral thesis was used by him again for the results of this 1857 paper.
- Riemann had quite a different opinion.
- We return at the end of this article to indicate how the problem of the use of Dirichlet's Principle in Riemann's work was sorted out.
- In 1858 Betti, Casorati and Brioschi visited Göttingen and Riemann discussed with them his ideas in topology.
- This gave Riemann particular pleasure and perhaps Betti in particular profited from his contacts with Riemann.
- These contacts were renewed when Riemann visited Betti in Italy in 1863.
- In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Göttingen on 30 July.
- A newly elected member of the Berlin Academy of Sciences had to report on their most recent research and Riemann sent a report on On the number of primes less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way.
- Riemann considered a very different question to the one Euler had considered, for he looked at the zeta function as a complex function rather than a real one.
- This is the famous Riemann hypothesis which remains today one of the most important of the unsolved problems of mathematics.
- Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function.
- Many of the results which Riemann obtained were later proved by Hadamard and de la Vallée Poussin.
- In the autumn of the year of his marriage Riemann caught a heavy cold which turned to tuberculosis.
- Riemann tried to fight the illness by going to the warmer climate of Italy.
- Having spent from August 1864 to October 1865 in northern Italy, Riemann returned to Göttingen for the winter of 1865-66, then returned to Selasca on the shores of Lake Maggiore on 16 June 1866.
- Finally let us return to Weierstrass's criticism of Riemann's use of the Dirichlet's Principle.
- This had the effect of making people doubt Riemann's methods.
- During the rest of the century Riemann's results exerted a tremendous influence: his way of thinking but little.
- Weierstrass firmly believed Riemann's results, despite his own discovery of the problem with the Dirichlet Principle.
- He asked his student Hermann Schwarz to try to find other proofs of Riemann's existence theorems which did not use the Dirichlet Principle.
- Klein, however, was fascinated by Riemann's geometric approach and he wrote a book in 1892 giving his version of Riemann's work yet written very much in the spirit of Riemann.
- Probably many took offence at its lack of rigour: Klein was too much in Riemann's image to be convincing to people who would not believe the latter.
- In 1901 Hilbert mended Riemann's approach by giving the correct form of Dirichlet's Principle needed to make Riemann's proofs rigorous.
- The search for a rigorous proof had not been a waste of time, however, since many important algebraic ideas were discovered by Clebsch, Gordan, Brill and Max Noether while they tried to prove Riemann's results.

Born 17 September 1826, Breselenz, Hanover (now Germany). Died 20 July 1866, Selasca, Italy.

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Analysis, Geometry, Knot Theory, Origin Germany, Number Theory, Physics, Special Numbers And Numerals, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive