Georg Friedrich Bernhard Riemann (; German: [ˈɡeːɔʁkˈfʁiːdʁɪçˈbɛʁnhaʁtˈʁiːman] ; 17September 1826 – 20July 1866) was a prominent German mathematician who significantly advanced the fields of analysis, number theory, and differential geometry. Within real analysis, his most notable achievements include the initial rigorous formulation of the integral, now known as the Riemann integral, and his extensive work on Fourier series. In complex analysis, he is particularly recognized for introducing Riemann surfaces, which pioneered a natural, geometric approach to the subject. His seminal 1859 publication concerning the prime-counting function, which presented the initial formulation of the Riemann hypothesis, stands as a cornerstone of analytic number theory. Riemann's groundbreaking work in differential geometry established the mathematical underpinnings for the theory of general relativity. He is widely regarded as one of history's most influential mathematicians.
Early Life
Born on September 17, 1826, Riemann originated from Breselenz, a village situated near Dannenberg within the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, served as an impoverished Lutheran pastor in Breselenz and was a veteran of the Napoleonic Wars. His mother, Charlotte Ebell, passed away in 1846. He was the second of six children. From an early age, Riemann displayed extraordinary mathematical aptitude, particularly in computational skills, yet he contended with profound timidity, glossophobia, and delicate health.
Academic Pursuits
In 1840, Riemann relocated to Hanover to reside with his grandmother and enroll in a lyceum, as this educational institution was unavailable in his native village. Following his grandmother's demise in 1842, he transferred to the Johanneum Lüneburg, a secondary school located in Lüneburg. While there, Riemann engaged in intensive biblical study, though his focus frequently shifted towards mathematics. His instructors were astonished by his capacity for complex mathematical computations, often surpassing their own expertise. At the age of 19, in 1846, he commenced studies in philology and Christian theology, intending to become a pastor and contribute to his family's financial stability.
In the spring of 1846, after his father secured sufficient funds, Riemann was sent to the University of Göttingen with the intention of pursuing a theology degree. Nevertheless, upon arrival, he commenced mathematical studies under Carl Friedrich Gauss, particularly attending lectures on the method of least squares. Gauss subsequently advised Riemann to abandon theology for mathematics; with his father's consent, Riemann transferred to the University of Berlin in 1847. During his tenure there, notable faculty included Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein. After two years in Berlin, he returned to Göttingen in 1849.
Academic Career
In 1854, Riemann delivered his inaugural lectures, which established the foundational principles of Riemannian geometry, thereby laying the groundwork for Albert Einstein's general theory of relativity. An endeavor to elevate Riemann to the position of extraordinary professor at the University of Göttingen occurred in 1857. While this promotion was unsuccessful, it did secure him a consistent salary. Subsequently, in 1859, upon the passing of Dirichlet, who occupied Gauss's esteemed chair at the University of Göttingen, Riemann was appointed to lead the university's mathematics department. Furthermore, he was the first to propose the utilization of dimensions beyond three or four for the description of physical reality.
In 1862, he married Elise Koch, and they subsequently had a daughter.
Later Life and Demise
In 1866, Riemann departed Göttingen amidst the conflict between the armies of Hanover and Prussia. He succumbed to tuberculosis during his third Italian journey, passing away in Selasca, currently a hamlet of Verbania on Lake Maggiore, where he was interred in the Biganzolo cemetery (Verbania).
Riemann was a devout Christian, the son of a Protestant minister, and viewed his mathematical pursuits as a form of divine service. He maintained a steadfast Christian faith throughout his life, regarding it as the paramount element of his existence. He passed away while reciting the Lord's Prayer with his wife, before its completion. Concurrently, in Göttingen, his housekeeper inadvertently disposed of numerous papers from his study, encompassing a significant volume of unpublished material. Given Riemann's reluctance to publish unfinished work, it is plausible that some profound insights were irretrievably lost.
Riemannian geometry
Riemann's published works pioneered new research domains at the intersection of analysis and geometry. These foundational contributions later evolved into central tenets of Riemannian geometry, algebraic geometry, and complex manifold theory. The conceptual framework of Riemann surfaces was further developed by Felix Klein and, notably, Adolf Hurwitz. This mathematical discipline constitutes a fundamental component of topology and continues to find innovative applications in mathematical physics.
In 1853, Gauss commissioned his student, Riemann, to compose a Habilitationsschrift addressing the fundamental principles of geometry. Riemann dedicated several months to formulating his theory of higher dimensions, culminating in a lecture delivered at Göttingen on June 10, 1854, titled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. This seminal work remained unpublished until 1868, twelve years later, when it was released by Dedekind, two years following Riemann's demise. Although its initial reception was reportedly subdued, it is now universally acknowledged as one of the most significant contributions to the field of geometry.
This foundational treatise established the discipline known as Riemannian geometry. Riemann successfully devised a method to generalize the differential geometry of surfaces—a concept Gauss himself elucidated in his theorema egregium—to n dimensions. Key components of this framework include the Riemannian metric and the Riemann curvature tensor. In the two-dimensional case of a surface, the curvature at any given point can be simplified to a scalar value, where surfaces exhibiting constant positive or negative curvature serve as exemplars of non-Euclidean geometries.
The Riemann metric, a tensor comprising a set of numbers at each spatial point, facilitates the measurement of speed along any trajectory; its integral yields the distance between the trajectory's terminal points. For instance, Riemann demonstrated that in a four-dimensional spatial context, ten distinct numerical values are required at each point to characterize distances and curvatures on a manifold, irrespective of its deformation.
Complex analysis
Within his dissertation, Riemann laid a geometric groundwork for complex analysis utilizing Riemann surfaces, thereby transforming multi-valued functions—such as the logarithm (characterized by infinitely many sheets) or the square root (with two sheets)—into single-valued functions. On these surfaces, complex functions manifest as harmonic functions (i.e., they adhere to Laplace's equation and consequently the Cauchy–Riemann equations), their properties defined by the positions of their singularities and the inherent topology of the surfaces. The topological genus of Riemann surfaces is mathematically expressed as , where the surface comprises leaves converging at branch points. When , the Riemann surface possesses parameters, known as moduli.
His contributions to this domain are extensive. The renowned Riemann mapping theorem posits that any simply connected domain within the complex plane is biholomorphically equivalent—meaning there exists a holomorphic bijection with a holomorphic inverse—to either or the interior of the unit circle. This theorem's generalization to Riemann surfaces is known as the uniformization theorem, a significant result established in the 19th century by Henri Poincaré and Felix Klein. Similarly, rigorous proofs for this generalization emerged only after the development of more sophisticated mathematical tools, specifically topology. In demonstrating the existence of functions on Riemann surfaces, Riemann employed a minimality condition, which he termed the Dirichlet principle. However, Karl Weierstrass identified a critical flaw in this proof: Riemann had overlooked the potential invalidity of his underlying assumption regarding the existence of a minimum, as the function space might lack completeness, thereby precluding a guaranteed minimum. Ultimately, the Dirichlet principle was rigorously established through David Hilbert's subsequent work in the Calculus of Variations. Despite this, Weierstrass held Riemann in high regard, particularly admiring his theory of abelian functions. Upon the publication of Riemann's work, Weierstrass consequently withdrew his own paper from Crelle's Journal, opting not to publish it. A strong mutual understanding developed between them during Riemann's Weierstrass subsequently encouraged his student, Hermann Amandus Schwarz, to develop alternative approaches to the Dirichlet principle within complex analysis, an endeavor in which Schwarz achieved success. An anecdote recounted by Arnold Sommerfeld illustrates the challenges contemporary mathematicians faced in comprehending Riemann's novel concepts. In 1870, Weierstrass reportedly took Riemann's dissertation on a holiday to Rigi, expressing difficulty in its comprehension. The physicist Hermann von Helmholtz assisted him overnight, subsequently remarking that the work was both "natural" and "very understandable".
Further significant contributions encompass his research on abelian functions and theta functions, particularly in the context of Riemann surfaces. Since 1857, Riemann had been engaged in a competitive endeavor with Weierstrass to resolve the Jacobian inverse problems for abelian integrals, which represent a generalization of elliptic integrals. Riemann approached this by employing theta functions of multiple variables, thereby reducing the problem to identifying the zeros of these functions. He also investigated period matrices, characterizing them via the "Riemannian period relations," which stipulate symmetry and a negative real part. Ferdinand Georg Frobenius and Solomon Lefschetz later demonstrated that the validity of this relation is equivalent to the embedding of —where denotes the lattice of the period matrix—into a projective space using theta functions. For specific values of , this construction yields the Jacobian variety of the Riemann surface, which exemplifies an abelian manifold.
Numerous mathematicians, including Alfred Clebsch, subsequently advanced Riemann's foundational work on algebraic curves. These theoretical frameworks were predicated upon the properties of functions defined over Riemann surfaces. For instance, the Riemann–Roch theorem—named in part after Roch, a student of Riemann—delineates the number of linearly independent differentials on a Riemann surface, subject to specified conditions regarding their zeros and poles.
Detlef Laugwitz posits that automorphic functions initially emerged in an essay concerning the Laplace equation applied to electrically charged cylinders. However, Riemann himself employed these functions for conformal mappings—for example, transforming topological triangles to a circle—in his 1859 lecture on hypergeometric functions and in his treatise on minimal surfaces.
Real analysis
In real analysis, Riemann introduced the Riemann integral during his habilitation, demonstrating that all piecewise continuous functions are integrable. The Stieltjes integral is also attributed to the Göttingen mathematician, leading to their combined designation as the Riemann–Stieltjes integral.
In his habilitation thesis on Fourier series, building upon the work of his mentor Dirichlet, Riemann established that Riemann-integrable functions can be represented by Fourier series. While Dirichlet had demonstrated this for continuous, piecewise-differentiable functions (characterized by a countable number of non-differentiable points), Riemann extended this by providing an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a scenario not addressed by Dirichlet. Furthermore, he proved the Riemann–Lebesgue lemma, which states that if a function is representable by a Fourier series, its Fourier coefficients approach zero as n becomes large.
Riemann's seminal essay also served as the foundational basis for Georg Cantor's investigations into Fourier series, which subsequently catalyzed the development of set theory.
In 1857, Riemann applied complex analytical methods to hypergeometric differential equations, illustrating their solutions through the behavior of closed paths around singularities, characterized by the monodromy matrix. The demonstration of the existence of such differential equations, given pre-defined monodromy matrices, constitutes one of Hilbert's problems.
Number Theory
Riemann significantly contributed to modern analytic number theory. In his sole, concise publication on number theory, he explored the zeta function, now named after him, thereby establishing its critical role in comprehending the distribution of prime numbers. The Riemann hypothesis emerged as one of several conjectures he proposed regarding the function's characteristics.
Riemann's work encompasses numerous other notable advancements. He demonstrated the functional equation for the zeta function, a relationship previously identified by Leonhard Euler, which is underpinned by a theta function. By summing this approximation function over the non-trivial zeros located on the line with a real part of 1/2, he derived an exact, "explicit formula" for .
Riemann was aware of Pafnuty Chebyshev's research concerning the Prime Number Theorem, as Chebyshev had visited Dirichlet in 1852.
Publications
Riemann's published works include:
- 1851 – Foundations for a general theory of the functions of a variable complex quantity (Foundations for a General Theory of Functions of a Variable Complex Quantity), Inaugural dissertation, Göttingen, 1851.
- 1857 – Theorie der Abelschen Functionen (Theory of Abelian Functions), Journal for pure and applied mathematics, Vol. 54, pp. 101–155.
- 1859 – About the number of prime numbers under a given size (On the Number of Primes Less Than a Given Magnitude), in: Monthly reports of the Prussian Academy of Sciences. Berlin, November 1859, pp. 671ff. This work includes Riemann's conjecture. About the number of prime numbers under a given size. (Wikisource), Facsimile of the manuscript Archived 2016-03-03 at the Wayback Machine with Clay Mathematics.
- 1861 – Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae (Mathematical Commentary, in which an Attempt is Made to Answer a Question Proposed by the Illustrious Paris Academy), submitted to the Paris Academy for a prize competition.
- 1867 – On the representability of a function by a trigonometric series (On the Representability of a Function by a Trigonometric Series), from the Thirteenth Volume of the Proceedings of the Royal Society of Sciences in Göttingen.
- 1868 – About the hypotheses underlying geometry (On the Hypotheses which Lie at the Foundation of Geometry). Treatises of the Royal Society of Sciences, Göttingen 1868. An English translation, On the hypotheses which lie at the foundation of geometry, by W.K. Clifford, appeared in Nature 8 (1873): 183, and was subsequently reprinted in Clifford's Collected Mathematical Papers (London: MacMillan, 1882; New York: Chelsea, 1968). This work is also included in Ewald, William B., ed., 1996, "From Kant to Hilbert: A Source Book in the Foundations of Mathematics", 2 vols. (Oxford University Press: 652–61).
- 1876 – Bernhard Riemann's Collected Mathematical Works and Scientific Papers, edited by Heinrich Weber with the collaboration of Richard Dedekind, Leipzig, B. G. Teubner 1876, 2nd edition 1892, reprinted by Dover 1953 (with contributions by Max Noether and Wilhelm Wirtinger, Teubner 1902). Later editions include The Collected Works of Bernhard Riemann: The Complete German Texts. Edited by Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., 1953, 1981, 2017.
- 1876 – Gravity, Electricity, and Magnetism, Hannover: Karl Hattendorff.
- 1882 – Lectures on Partial Differential Equations, 3rd edition. Braunschweig 1882.
- 1901 – The Partial Differential Equations of Mathematical Physics According to Riemann's Lectures. Riemann, Bernhard (1901). Weber, Heinrich Martin (ed.). "Die partiellen differential-gleichungen der mathematischen physik nach Riemann's Vorlesungen". Friedrich Vieweg und Sohn.
- 2004 – Riemann, Bernhard (2004), Collected Papers, Kendrick Press, Heber City, UT, ISBN 978-0-9740427-2-5, MR 2121437List of entities named after Bernhard Riemann.
- List of things named after Bernhard Riemann
- Non-Euclidean geometry.
- On the Number of Primes Less Than a Given Magnitude, Riemann's 1859 paper introducing the complex zeta function.
References
Derbyshire, John (2003), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Washington, DC: John Henry Press, ISBN 0-309-08549-7.
- Derbyshire, John (2003), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Washington, DC: John Henry Press, ISBN 0-309-08549-7Monastyrsky, Michael (1999), Riemann, Topology and Physics, Boston, MA: Birkhäuser, ISBN 0-8176-3789-3Ji, Lizhen; Papadopoulos, Athanese; Yamada, Sumio, eds. (2017). From Riemann to Differential Geometry and Relativity. Springer. ISBN 9783319600390.Bernhard Riemann's academic lineage at the Mathematics Genealogy Project.
- Bernhard Riemann at the Mathematics Genealogy Project
- The collected mathematical papers of Georg Friedrich Bernhard Riemann.
- Riemann's publications.
- O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor History of Mathematics Archive, University of St AndrewsWeisstein, Eric Wolfgang (ed.). "Riemann, Bernhard (1826–1866)". ScienceWorld.