was an influential
GermanGermany , officially the Federal Republic of Germany , is a country in Central Europe. It is bordered to the north by the North Sea, Denmark, and the Baltic Sea; to the east by Poland and the Czech Republic; to the south by Austria and Switzerland; and to the west by France, Luxembourg, Belgium,...
mathematicianMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
who made contributions to
analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
and differential geometry, some of them enabling the later development of
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
.
Early life
Riemann was born in Breselenz, a village near
DannenbergDannenberg is a town in the district Lüchow-Dannenberg, in Lower Saxony, Germany. It is situated near the river Elbe, approx. 30 km north of Salzwedel, and 50 km south-east of Lüneburg...
in the
Kingdom of HanoverThe Kingdom of Hanover was established in October 1814 by the Congress of Vienna, with the restoration of George III to his Hanoverian territories after the Napoleonic era...
in what is
GermanyGermany , officially the Federal Republic of Germany , is a country in Central Europe. It is bordered to the north by the North Sea, Denmark, and the Baltic Sea; to the east by Poland and the Czech Republic; to the south by Austria and Switzerland; and to the west by France, Luxembourg, Belgium,...
today. His father, Friedrich Bernhard Riemann, was a poor
LutheranianLutheranism is a major branch of Western Christianity that identifies with the teachings of the 16th century German reformer Martin Luther. Luther's efforts to reform the theology and practice of the church launched the Protestant Reformation...
pastor in Breselenz who fought in the
Napoleonic WarsThe Napoleonic Wars were a series of conflicts declared against Napoleon's French Empire and changing sets of European allies by opposing coalitions that ran from 1803 to 1815. As a continuation of the wars sparked by the French Revolution of 1789, they revolutionized European armies and played...
. His mother died before her children were grown up. 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
BibleThe Bible contains the central religious texts of Judaism and Christianity. Modern Judaism generally recognizes a single set of canonical books known as the Tanakh, or Hebrew Bible, as it is written almost entirely in the Hebrew language, with some small portions in Aramaic...
intensively, but he was often distracted by mathematics. To this end, he even tried to
proveIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
mathematically the correctness of the Book of Genesis. His teachers were amazed by his adept ability to solve complicated mathematical operations, in which he often outstripped his instructor's knowledge. During 1840, Riemann went to
HanoverHanover or Hannover , on the river Leine, is the capital of the federal state of Lower Saxony , Germany and was once by personal union the family seat of the Hanoverian Kings of Great Britain, in their dignities as the dukes of Brunswick-Lüneburg Hanover or Hannover , on the river Leine, is...
to live with his grandmother and attend
lyceumA Lyceum can be*an educational institution , or*a public hall used for cultural events like concerts.*Mount 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
philologyPhilology considers both form and meaning in linguistic expression, combining linguistics and literary studies.Classical philology is the philology of the Greek, Latin and Sanskrit languages...
and
theologyThe term "theology" literally means the study of God, deriving from the Greek word theos, meaning 'God', and the suffix -ology from the Greek word logos meaning "discourse", "theory", or "reasoning"...
in order to become a priest and help with his family's finances.
During the spring of 1846, his father (Friedrich Riemann), after gathering enough money to send Riemann to university, allowed him to stop studying theology and start studying
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
. He was sent to the renowned University of Göttingen, where he first met
Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics...
, and attended his lectures on the method of least squares.
In 1847, Riemann moved to
BerlinBerlin is the capital city and one of sixteen states of Germany. With a population of 3.4 million within its city limits, Berlin is Germany's largest city. It is the second most populous city and the eighth most populous urban area in the European Union...
, where
JacobiCarl Gustav Jacob Jacobi was a Prussian mathematician, widely considered to be the most inspiring teacher of his time and one of the greatest mathematicians of all time.-Biography:...
,
DirichletJohann Peter Gustav Lejeune Dirichlet was a German mathematician credited with the modern formal definition of a function.- Biography :...
,
SteinerJakob Steiner was a Swiss mathematician.He was born in the village of Utzenstorf, Canton of Bern. At eighteen he became a pupil of Heinrich Pestalozzi, and afterwards studied at Heidelberg. Thence he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became...
, and Eisenstein 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 founded the field of
Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
and thereby set the stage for
EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
's general theory of 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
DirichletJohann Peter Gustav Lejeune Dirichlet was a German mathematician credited with the modern formal definition of a function.- Biography :...
's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality—an idea that was ultimately vindicated with Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.
Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of
tuberculosisTuberculosis is a common and often deadly infectious disease caused by mycobacteria...
during his third journey to
ItalyItaly , officially the Italian Republic , is a country located on the Italian Peninsula in Southern Europe and on the two largest islands in the Mediterranean Sea, Sicily and Sardinia. Italy shares its northern, Alpine boundary with France, Switzerland, Austria and Slovenia...
in Selasca (now a hamlet of
VerbaniaVerbania is a city and comune on the shore of Lake Maggiore, Piedmont in northwest Italy. It was created by the 1939 merger of the cities of Intra and Pallanza...
on
Lake MaggioreLake Maggiore is the most westerly of the three large prealpine lakes of Italy and the second largest after Lake Garda. It lies approximately at .It has a surface area of about 213 km², a maximum length of 54 km and, at its widest, is 12 km...
) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.
Influence
Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of
Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
,
algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
, and
complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
theory. The theory of
Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s was elaborated by
Felix KleinFelix Christian Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
and particularly
Adolf HurwitzAdolf Hurwitz , was a German mathematician, and was described by Jean-Pierre Serre as "one of the most important figures in mathematics in the second half of the nineteenth century".-Early life:...
. This area of mathematics is part of the foundation of
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
, and is still being applied in novel ways to
mathematical physicsMathematical physics is the scientific discipline concerned with the interface of mathematics and physics. The Journal of Mathematical Physics defines it as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the...
.
Riemann made major contributions to
real analysisReal analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
. He defined the
Riemann integralIn the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define...
by means of
Riemann sumIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation...
s, developed a theory of
trigonometric seriesIn mathematics, a trigonometric series is any series of the form:It is called a Fourier series when the terms and have the form:where is an integrable function....
that are not
Fourier seriesIn mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines . The study of Fourier series is a branch of Fourier analysis...
—a first step in
generalized functionIn mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognised theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
theory—and studied the
Riemann-Liouville differintegralIn mathematics, the Riemann–Liouville integral associates with a real function ƒ : R → R another function Iαƒ of the same kind for each value of the parameter α > 0...
.
He made some famous contributions to modern
analytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number-theoretical problems. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
. In a single short paper (the only one he published on the subject of number theory), he introduced the
Riemann zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann who introduced it in 1859, is a prominent function of great significance in number theory because of its relation to the distribution of prime numbers...
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 hypothesisIn mathematics, the Riemann hypothesis, proposed by , is a conjecture about the distribution of the zeros of the Riemann zeta-function stating that all non-trivial zeros of the Riemann zeta function have real part 1/2...
.
He applied the
Dirichlet principleIn mathematics, Dirichlet's principle in potential theory states that, if the function u is the solution to Poisson's equationon a domain of with boundary conditionthen u can be obtained as the minimizer of the Dirichlet's energy...
from variational calculus to great effect; this was later seen to be a powerful
heuristicHeuristic is an adjective for experience-based techniques that help in problem solving, learning and discovery. A heuristic method is particularly used to rapidly come to a solution that is hoped to be close to the best possible answer, or 'optimal solution'...
rather than a rigorous method. Its justification took at least a generation. His work on
monodromyIn mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
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 singularitiesIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
.
Euclidean geometry versus Riemannian geometry
In 1853,
GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics...
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 is 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 geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
. Riemann found the correct way to extend into
n dimensions the differential geometry of surfaces, which Gauss himself proved in his
theorema egregiumGauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...
. The fundamental object is called the
Riemann curvature tensorIn the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
. 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
spaceSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of the boundless four-dimensional...
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
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric.
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.
External links