Everything about Bernhard Riemann totally explained
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.
Biography
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 who fought in the
Napoleonic Wars. His mother died before her children were grown. Riemann was the second of six children, shy, and suffered from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as fantastic calculation abilities, from an early age, but suffered from timidity and a fear of speaking in public.
Middle life
In high school, Riemann studied the
Bible intensively, but his mind often drifted back to mathematics. He even tried to
prove mathematically the correctness of the
Book of Genesis. His teachers were amazed by his genius and his ability to solve extremely complicated mathematical operations. He often outstripped his instructor's knowledge. In 1840, Riemann went to
Hanover to live with his grandmother and attend
lyceum (middle school). After the death of his grandmother in 1842, he attended high school at the
Johanneum 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 (Friedrich Riemann), after gathering 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, Riemann 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
Bernhard 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. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the
University of Göttingen. Although this attempt failed, it did result in Riemann finally being 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 greatly 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).
Influence
Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become 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 is part of the foundation of
topology, and is still being applied in novel ways to
mathematical physics.
Riemann made major contributions to
real analysis. He defined the
Riemann integral by means of
Riemann sums, 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 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. 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.
Euclidean geometry versus Riemannian geometry
In 1853,
Gauss asked his student Riemann 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 at Göttingen in 1854, the mathematical public received it with enthusiasm, and it's one of the most important works in geometry. It was titled
Über die Hypothesen welche der Geometrie zu Grunde liegen (loosely: "On the foundations of geometry"; more precisely, "On the hypotheses which underlie geometry"), and was published in 1868.
The subject founded by this work is
Riemannian geometry. Riemann found the correct way to extend into
n dimensions the
differential geometry of surfaces, which Gauss himself proved in his
theorema egregium. The fundamental object is 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.
Further Information
Get more info on 'Bernhard Riemann'.
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