Bernhard Riemann

Overview

**Georg Friedrich Bernhard Riemann**ˈʁiːman (September 17, 1826 – July 20, 1866) was an influential German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

who made lasting contributions to analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

and differential geometry, some of them enabling the later development of general relativity

General relativity

General 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...

.

Riemann was born in Breselenz, a village near Dannenberg

Dannenberg (Elbe)

Dannenberg 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 Hanover

Kingdom of Hanover

The 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. It succeeded the former Electorate of Brunswick-Lüneburg , and joined with 38 other sovereign states in the German...

in what is the Federal Republic of Germany

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

today. His father, Friedrich Bernhard Riemann, was a poor Lutheran

Lutheranism

Lutheranism is a major branch of Western Christianity that identifies with the theology of Martin Luther, a German reformer. Luther's efforts to reform the theology and practice of the church launched the Protestant Reformation...

pastor in Breselenz who fought in the Napoleonic Wars

Napoleonic Wars

The Napoleonic Wars were a series of wars declared against Napoleon's French Empire by opposing coalitions that ran from 1803 to 1815. As a continuation of the wars sparked by the French Revolution of 1789, they revolutionised European armies and played out on an unprecedented scale, mainly due to...

.

Unanswered Questions

Encyclopedia

**Georg Friedrich Bernhard Riemann**ˈʁiːman (September 17, 1826 – July 20, 1866) was an influential German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

who made lasting contributions to analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

and differential geometry, some of them enabling the later development of general relativity

General relativity

General 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...

.

### Early years

Riemann was born in Breselenz, a village near DannenbergDannenberg (Elbe)

Dannenberg 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 Hanover

Kingdom of Hanover

The 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. It succeeded the former Electorate of Brunswick-Lüneburg , and joined with 38 other sovereign states in the German...

in what is the Federal Republic of Germany

Germany

today. His father, Friedrich Bernhard Riemann, was a poor Lutheran

Lutheranism

Lutheranism is a major branch of Western Christianity that identifies with the theology of Martin Luther, a German reformer. Luther's efforts to reform the theology and practice of the church launched the Protestant Reformation...

pastor in Breselenz who fought in the Napoleonic Wars

Napoleonic Wars

The Napoleonic Wars were a series of wars declared against Napoleon's French Empire by opposing coalitions that ran from 1803 to 1815. As a continuation of the wars sparked by the French Revolution of 1789, they revolutionised European armies and played out on an unprecedented scale, mainly due to...

. His mother, Charlotte Ebell, died before her children had reached adulthood. 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.

### Education

During 1840, Riemann went to HanoverHanover

Hanover 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, under their title as the dukes of Brunswick-Lüneburg...

to live with his grandmother and attend lyceum

Lyceum

The lyceum is a category of educational institution defined within the education system of many countries, mainly in Europe. The definition varies between countries; usually it is a type of secondary school.-History:...

(middle school). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg.

In high school, Riemann studied the Bible

Bible

The Bible refers to any one of the collections of the primary religious texts of Judaism and Christianity. There is no common version of the Bible, as the individual books , their contents and their order vary among denominations...

intensively, but he was often distracted by mathematics. His teachers were amazed by his adept ability to solve complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying philology

Philology

Philology is the study of language in written historical sources; it is a combination of literary studies, history and linguistics.Classical philology is the philology of Greek and Classical Latin...

and theology

Theology

Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

in order to become a priest and help with his family's finances.

During the spring of 1846, his father, after gathering enough money, sent Riemann to university at the renowned University of Göttingen, where he planned to study towards a degree in Theology. However, once there, he began studying mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

under Carl Friedrich Gauss

Carl Friedrich Gauss

Johann 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.Sometimes referred to as the Princeps mathematicorum...

(specifically his lectures on the method of least squares). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his parents' approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Jacobi

Carl Gustav Jakob Jacobi

Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...

, Dirichlet

Johann Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...

, Steiner

Jakob Steiner

Jakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...

, and Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.

### Academia

Riemann held his first lectures in 1854, which founded the field of Riemannian geometryRiemannian geometry

Riemannian 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 Einstein

Albert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

'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 Dirichlet

Johann Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...

'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

General relativity

General 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...

in the early 20th century. In 1862 he married Elise Koch and had a daughter.

### Austro-Prussian War

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosisTuberculosis

Tuberculosis, MTB, or TB is a common, and in many cases lethal, infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis. Tuberculosis usually attacks the lungs but can also affect other parts of the body...

during his third journey to Italy

Italy

Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

in Selasca (now a hamlet of Verbania

Verbania

Verbania is a city and comune on the shore of Lake Maggiore, Piedmont, in northwest Italy, about north-west of Milan and about from Locarno in Switzerland.-Overview:...

on Lake Maggiore

Lake Maggiore

Lake Maggiore is a large lake located on the south side of the Alps. It is the second largest of Italy and largest of southern Switzerland. Lake Maggiore is the most westerly of the three great prealpine lakes of Italy, it extends for about 70 km between Locarno and Arona.The climate is mild...

) 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

Riemannian 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 geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which 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 diverse fields as complex...

, and complex manifold

Complex manifold

In 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 surface

Riemann surface

In 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 Klein

Felix Klein

Christian Felix 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 Hurwitz

Adolf Hurwitz

Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...

. This area of mathematics is part of the foundation of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, and is still being applied in novel ways to mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

.

Riemann made major contributions to real analysis

Real analysis

Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...

. He defined the Riemann integral

Riemann integral

In 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. The Riemann integral is unsuitable for many theoretical purposes...

by means of Riemann sum

Riemann sum

In 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. The method was named after German mathematician Bernhard Riemann....

s, developed a theory of trigonometric series that are not Fourier series

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

—a first step in generalized function

Generalized function

In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized 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 differintegral.

He made some famous contributions to modern analytic number theory

Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. 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

On the Number of Primes Less Than a Given Magnitude

die Anzahl der Primzahlen unter einer gegebenen is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.Although it is the only paper he ever published on number theory, it...

(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

Riemann hypothesis

In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

.

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful heuristic

Heuristic

Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

rather than a rigorous method. Its justification took at least a generation. His work on monodromy

Monodromy

In 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 singularities*

.

Mathematical singularity

In 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 GaussCarl Friedrich Gauss

Johann 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.Sometimes referred to as the Princeps mathematicorum...

asked his student Riemann to prepare a

*Habilitationsschrift*on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen in 1854 entitled

*Über die Hypothesen welche der Geometrie zu Grunde liegen*("On the hypotheses which underlie geometry"). When it was finally published in 1868, two years after his death, the mathematical public received it with enthusiasm and it is now recognized as one of the most important works in geometry.

The subject founded by this work is Riemannian geometry

Riemannian geometry

Riemannian 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 egregium*

. The fundamental object is called the Riemann curvature tensor

Theorema Egregium

Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...

Riemann curvature tensor

In 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 space (i.e., a tensorTensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

) which 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

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific 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” translated by W.K.Clifford, Nature 8 1873 183- reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea).
- 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.

## See also

- List of topics named after Bernhard Riemann
- Riemann's 1859 paper introducing the complex zeta functionOn the Number of Primes Less Than a Given Magnitudedie Anzahl der Primzahlen unter einer gegebenen is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.Although it is the only paper he ever published on number theory, it...

## External links

- The Mathematical Papers of Georg Friedrich Bernhard Riemann
- All publications of Riemann can be found at: http://www.emis.de/classics/Riemann/
- Bernhard Riemann – one of the most important mathematicians
- Bernhard Riemann's inaugural lecture