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Bernhard Riemanns father, Friedrich Bernhard Riemann, was a Lutheran minister. Friedrich Riemann married Charlotte Ebell when he was in his middle age. ... Friedrich Riemann acted as teacher to his children and he taught Bernhard until he was ten years old. ...
In the spring of 1846 Riemann enrolled at the University of Göttingen. ... Riemann was always very close to his family and he would never have changed courses without his fathers permission. 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 Riemanns genius. Stern, however, certainly did realise that he had a remarkable student and later described Riemann at this time saying that he:-
. ...
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. Klein writes in [4]:-
Riemann was bound to Dirichlet by the strong inner sympathy of a like mode of thought. ... His manner suited Riemann, who adopted it and worked according to Dirichlets methods. ... 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.
Riemanns thesis studied the theory of complex variables and, in particular, what we now call Riemann 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 Dirichlets lectures in Berlin. ... In his report on the thesis Gauss described Riemann as having:-
. ...
On Gausss 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 Riemanns expectations, Gauss chose the lecture on geometry. ... In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras theorem.
Approximate Word count = 2639
Approximate Pages = 10.6
(250 words per page double spaced)
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