Riemann's essay made considerable progress on this problem, first by giving a criterion for a function to be integrable or as we now say, Riemann integrable , and then by obtaining a necessary condition for a Riemann integrable function to be representable by a Fourier series.
For his Habiltationsvortrag Riemann proposed three topics, and against his expectations Gauss chose the one on geometry. Riemann's lecture, "On the hypotheses that lie at the foundation of geometry" was given on June 10, This extraordinary work introduced what is now called an n -dimensional Riemannian manifold and its curvature tensor.
It also, prophetically, discussed the relation of this mathematical space to actual space. Riemann's vision was realized by Einstein's general theory of relativity sixty years later. Probably the only person in the audience who appreciated the depth of Riemann's work was Gauss, who had done pioneering work in differential geometry.
A report of Riemann's lecture was not published until , after his death. Although a Privatdozent could collect fees from students, the position had no salary. With Dirichlet's help, Riemann obtained a small paid post. He did not become an assistant professor until , which was also the year he published his researches on Abelian functions.
Abelian functions were studied by Abel and Jacobi ; they are a generalization of elliptic functions. He had five siblings and was the second of the six. His mother died before he reached his adulthood and also suffered multiple nervous breakdowns as a child. He was an extremely timid and shy child who perpetually suffered from social anxiety. At the same time he displayed incredible aptitude for mathematics. He was seen studying Bible rigorously in high school, simultaneously he had shown increasing interest in mathematics.
His teachers were impressed by his proficient skills in solving complex mathematical problems. Beginning in , he attended the Gymnasium in Hanover, where he lived with his grandmother.
In , when his grandmother died, he moved to the Gymnasium at Luneburg. At the age of 19, Bernhard entered the University of Gottingen, where he studied philosophy and theology.
However, seeing his interest in mathematics, his teacher, Carl Friedrich Gauss, encouraged him to talk to his parents about switching to a mathematics degree rather than theology. After his father agreed to that, he was transferred to the University of Berlin in In , after the death of Guass who was the appointed professor of physics at the University of Gottingen , Riemann was appointed extraordinary professor at the university. During that time, he published his paper on the theory of Abelian functions.
Bernhard lived a relatively short life, but he made great contributions during his lifetime. Although he published only a few papers, his name is attached to a variety of topics in several branches of mathematics, such as Riemannian geometry, the Cauchy-Riemann equation, Riemann surfaces, the Riemannian geometry and Riemann Hypothesis. This paper, which was published in , marked the beginning of analytic number theory where arithmetical objects are studied by analytical means.
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. 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. It therefore introduced topological methods into complex function theory.
The work builds on Cauchy 's foundations of the theory of complex variables built up over many years and also on Puiseux 's ideas of branch points. 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. The Dirichlet Principle did not originate with Dirichlet , however, as Gauss , Green and Thomson had all made use if it. Riemann's thesis, one of the most remarkable pieces of original work to appear in a doctoral thesis, was examined on 16 December In his report on the thesis Gauss described Riemann as having He spent thirty months working on his Habilitation dissertation which was on the representability of functions by trigonometric series.
He gave the conditions of a function to have an integral, what we now call the condition of Riemann integrability. In the second part of the dissertation he examined the problem which he described in these words:- While preceding papers have shown that if a function possesses such and such a property, then it can be represented by a Fourier series , we pose the reverse question: if a function can be represented by a trigonometric series, what can one say about its behaviour.
To complete his Habilitation Riemann had to give a lecture. He prepared three lectures, two on electricity and one on geometry. Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry. There were two parts to Riemann's lecture. In the first part he posed the problem of how to define an n n n -dimensional space and ended up giving a definition of what today we call a Riemannian space.
Freudenthal writes in [ 1 ] :- It possesses shortest lines, now called geodesics, which resemble ordinary straight lines. 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 's theorem. In fact 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.
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