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35-652: Riemann was born on 17 September 1826 in Breselenz , a village near Dannenberg in the Kingdom of Hanover . His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars . His mother, Charlotte Ebell, died in 1846. Riemann was the second of six children. Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and

70-416: A manifold , no matter how distorted it is. In his dissertation, he established a geometric foundation for complex analysis through Riemann surfaces , through which multi-valued functions like the logarithm (with infinitely many sheets) or the square root (with two sheets) could become one-to-one functions . Complex functions are harmonic functions (that is, they satisfy Laplace's equation and thus

105-633: A Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma : if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large  n . Riemann's essay was also the starting point for Georg Cantor 's work with Fourier series, which was the impetus for set theory . He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented

140-524: A competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of elliptic integrals . Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated period matrices and characterized them through the "Riemannian period relations" (symmetric, real part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz

175-739: A fear of speaking in public. During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years), because such a type of school was not accessible from his home village. After the death of his grandmother in 1842, he transferred to the Johanneum Lüneburg , a high school in Lüneburg . There, Riemann studied the Bible intensively, but he was often distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at

210-455: A hamlet of Verbania on Lake Maggiore ), where he was buried in the cemetery in Biganzolo (Verbania). Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting

245-478: A number (scalar), with the surfaces of constant positive or negative curvature being models of the non-Euclidean geometries . The Riemann metric is a collection of numbers at every point in space (i.e., a tensor ) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on

280-483: A theta function lies. Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for π ( x ) {\displaystyle \pi (x)} . Riemann knew of Pafnuty Chebyshev 's work on the Prime Number Theorem . He had visited Dirichlet in 1852. Riemann's works include: Breselenz Jameln

315-746: Is a municipality in the district Lüchow-Dannenberg , in Lower Saxony , Germany . Jameln is part of the Samtgemeinde ("collective municipality") Elbtalaue . The main village in the municipality is Jameln, with around 450 inhabitants. Since the Gebietsreform  [ de ] (municipality reform) in 1972, Jameln consists of ten villages: Additionally, there are three hamlets ( Wohnplätze  [ de ] ): Before 1972, Hoheluft belonged to Volkfien, Jamelner Mühle to Jameln and Krammühle to Breselenz. Mathematician Bernhard Riemann (1826–1866)

350-515: Is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below). The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about. In all of

385-584: Is the Jacobian variety of the Riemann surface, an example of an abelian manifold. Many mathematicians such as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on

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420-578: The Cauchy–Riemann equations ) on these surfaces and are described by the location of their singularities and the topology of the surfaces. The topological "genus" of the Riemann surfaces is given by g = w / 2 − n + 1 {\displaystyle g=w/2-n+1} , where the surface has n {\displaystyle n} leaves coming together at w {\displaystyle w} branch points. For g > 1 {\displaystyle g>1}

455-563: The Dirichlet principle . Karl Weierstrass found a gap in the proof: Riemann had not noticed that his working assumption (that the minimum existed) might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed. Through the work of David Hilbert in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass

490-822: The Lord's Prayer with his wife and died before they finished saying the prayer. Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost. Riemann's tombstone in Biganzolo (Italy) refers to Romans 8:28 : Georg Friedrich Bernhard Riemann Professor in Göttingen born in Breselenz, 17 September 1826 died in Selasca, 20 July 1866 Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of

525-594: The method of least squares ). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Carl Gustav Jacob Jacobi , Peter Gustav Lejeune Dirichlet , Jakob Steiner , and Gotthold Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded

560-401: The zeta function that now bears his name, establishing its importance for understanding the distribution of prime numbers . The Riemann hypothesis was one of a series of conjectures he made about the function's properties. In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler ), behind which

595-420: The 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry . Every smooth manifold admits a Riemannian metric , which often helps to solve problems of differential topology . It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds , which (in four dimensions) are

630-476: The Riemann surface has ( 3 g − 3 ) {\displaystyle (3g-3)} parameters (the " moduli "). His contributions to this area are numerous. The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either C {\displaystyle \mathbb {C} } or to

665-548: The University of Göttingen), he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality. In 1862 he married Elise Koch; their daughter Ida Schilling was born on 22 December 1862. Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now

700-424: The age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen , where he planned to study towards a degree in theology . However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on

735-456: The behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein 's general theory of relativity , made profound impact on group theory and representation theory , as well as analysis , and spurred the development of algebraic and differential topology . Riemannian geometry was first put forward in generality by Bernhard Riemann in

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770-399: The field of Riemannian geometry and thereby set the stage for Albert Einstein '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 the death of Dirichlet (who held Gauss 's chair at

805-870: The field of real analysis , he discovered the Riemann integral in his habilitation . Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, and so they are named together the Riemann–Stieltjes integral . In his habilitation work on Fourier series , where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of

840-402: The foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen on 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen . It was not published until twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow, but it is now recognized as one of

875-436: The interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous uniformization theorem , which was proved in the 19th century by Henri Poincaré and Felix Klein . Here, too, rigorous proofs were first given after the development of richer mathematical tools (in this case, topology). For the proof of the existence of functions on Riemann surfaces, he used a minimality condition, which he called

910-400: The main objects of the theory of general relativity . Other generalizations of Riemannian geometry include Finsler geometry . There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature. The following articles provide some useful introductory material: What follows

945-489: The most important works in geometry. 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 objects are called the Riemannian metric and the Riemann curvature tensor . For the surface (two-dimensional) case, the curvature at each point can be reduced to

980-468: The solutions through the behaviour of closed paths about singularities (described by the monodromy matrix ). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems. Riemann 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 investigated

1015-411: 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 . In 1853, Gauss asked Riemann, his student, to prepare a Habilitationsschrift on

1050-402: The validity of this relation is equivalent with the embedding of C n / Ω {\displaystyle \mathbb {C} ^{n}/\Omega } (where Ω {\displaystyle \Omega } is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of n {\displaystyle n} , this

1085-455: The vision of Bernhard Riemann expressed in his inaugural lecture " Ueber die Hypothesen, welche der Geometrie zu Grunde liegen " ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R . Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and

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1120-456: The zeros and poles) of a Riemann surface. According to Detlef Laugwitz , automorphic functions appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces . In

1155-651: Was born in Breselenz. This Lüchow-Dannenberg location article is a stub . You can help Misplaced Pages by expanding it . Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds , defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle , length of curves , surface area and volume . From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with

1190-542: Was successful. An anecdote from Arnold Sommerfeld shows the difficulties which contemporary mathematicians had with Riemann's new ideas. In 1870, Weierstrass had taken Riemann's dissertation with him on a holiday to Rigi and complained that it was hard to understand. The physicist Hermann von Helmholtz assisted him in the work overnight and returned with the comment that it was "natural" and "very understandable". Other highlights include his work on abelian functions and theta functions on Riemann surfaces. Riemann had been in

1225-410: Was very impressed with Riemann, especially with his theory of abelian functions . When Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859. Weierstrass encouraged his student Hermann Amandus Schwarz to find alternatives to the Dirichlet principle in complex analysis, in which he

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