History of manifolds and varieties
Encyclopedia
The study of manifold
s combines many important areas of mathematics
: it generalizes concepts such as curve
s and surfaces as well as ideas from linear algebra
and topology
. Certain special classes of manifolds also have additional algebraic structure; they may behave like group
s, for instance. In that case, they are called Lie Group
s. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic group
s.
The term "manifold" comes from German Mannigfaltigkeit, by Riemann.
In Romance languages, this is translated as "variety" – such spaces with a differentiable structure are called "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties".
In English, "manifold" refers to spaces with a differentiable or topological structure,
while "variety" refers to spaces with an algebraic structure, as in algebraic varieties.
, which considers spaces where Euclid
's parallel postulate
fails. Saccheri first studied this geometry in 1733. Lobachevsky
, Bolyai
, and Riemann
developed the subject further 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space
; these are called hyperbolic geometry
and elliptic geometry
. In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positive curvature
, respectively.
Carl Friedrich Gauss
may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium
gives a method for computing the curvature
of a surface
without considering the ambient space
in which the surface lies. In modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space.
Another, more topological
example of an intrinsic property
of a manifold is the Euler characteristic
. For a non-intersecting graph
in the Euclidean plane
, with V vertices (or corners), E edges and F faces (counting the exterior) Euler
showed that V-E+F= 2. Thus 2 is called the Euler characteristic of the plane. By contrast, in 1813 Antoine-Jean Lhuilier showed that the Euler characteristic of the torus
is 0, since the complete graph
on seven points can be embedded into the torus. The Euler characteristic of other surfaces is a useful topological invariant, which has been extended to higher dimensions using Betti number
s. In the mid nineteenth century, the Gauss–Bonnet theorem
linked the Euler characteristic to the Gaussian curvature.
Lagrangian mechanics
and Hamiltonian mechanics
, when considered geometrically, are naturally manifold theories. All these use the notion of several characteristic axes or dimension
s (known as generalized coordinates
in the latter two cases), but these dimensions do not lie along the physical dimensions of width, height, and breadth.
In the early 19th century the theory of elliptic function
s succeeded in giving a basis for the theory of elliptic integral
s, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square root
s of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen?
In the work of Niels Abel and Carl Jacobi
, the answer was formulated: the resulting integral would involve functions of two complex variables, having four independent periods (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an abelian surface): what would now be called the Jacobian
of a hyperelliptic curve of genus 2.
was the first to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German
term, Mannigfaltigkeit, which William Kingdon Clifford
translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction
, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifold
s and Riemann surface
s are named after Bernhard Riemann.
In 1857, Riemann introduced the concept of Riemann surface
s as part of a study of the process of analytic continuation
; Riemann surfaces are now recognized as one-dimensional complex manifolds. He also furthered the study of abelian and other multi-variable complex functions.
, inventor of the word "topology
", wrote an 1847 paper "Vorstudien zur Topologie" in which he defined a "complex
". He first defined the Möbius strip
in 1861 (rediscovered four years later by Möbius
), as an example of a non-orientable surface.
After Abel, Jacobi, and Riemann, some of the most important contributors to the theory of abelian functions were Weierstrass
, Frobenius
, Poincaré
and Picard
. The subject was very popular at the time, already having a large literature. By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions.
's 1895 paper Analysis Situs studied three-and-higher-dimensional manifolds, giving rigorous definitions of homology, homotopy (which had originally been defined in the context of late nineteenth-century knot theory
, developed by Maxwell
and others), and Betti number
s and raised a question, today known as the Poincaré conjecture
, based his new concept of the fundamental group
. As of 2006, a consensus among experts is that recent work by Grigori Perelman may have answered this question, after nearly a century of effort by many mathematicians.
gave an intrinsic definition for differentiable manifolds in 1912. During the 1930s Hassler Whitney
and others clarified the foundational
aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group
theory.
The Whitney embedding theorem, which showed that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as in the extrinsic definition, showing that the two concepts of manifold were equivalent. Due to this unification, it is said to be the first complete exposition of the modern concept of manifold.
Eventually, in the 1920s
, Lefschetz
laid the basis for the study of abelian functions in terms of complex tori. He also appears to have been the first to use the name "abelian variety
"; in Romance languages
, "variety" was used to translate Riemann's term "Mannigfaltigkeit". It was Weil
in the 1940s
who gave this subject its modern foundations in the language of algebraic geometry.
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....
s combines many important areas of 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...
: it generalizes concepts such as curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s and surfaces as well as ideas from linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and 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...
. Certain special classes of manifolds also have additional algebraic structure; they may behave like group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s, for instance. In that case, they are called Lie Group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s.
The term "manifold" comes from German Mannigfaltigkeit, by Riemann.
In Romance languages, this is translated as "variety" – such spaces with a differentiable structure are called "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties".
In English, "manifold" refers to spaces with a differentiable or topological structure,
while "variety" refers to spaces with an algebraic structure, as in algebraic varieties.
Background
Ancestral to the modern concept of a manifold were several important results of 18th and 19th century mathematics. The oldest of these was Non-Euclidean geometryNon-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
, which considers spaces where Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
fails. Saccheri first studied this geometry in 1733. Lobachevsky
Nikolai Ivanovich Lobachevsky
Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, renowned primarily for his pioneering works on hyperbolic geometry, otherwise known as Lobachevskian geometry...
, Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...
, and Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
developed the subject further 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
; these are called hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
and elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
. In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positive curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
, respectively.
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...
may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium
Theorema Egregium
Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...
gives a method for computing the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
without considering the ambient space
Ambient space
An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object.-Mathematics:In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object...
in which the surface lies. In modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space.
Another, more topological
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...
example of an intrinsic property
Topological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic...
of a manifold is the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
. For a non-intersecting graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
in the Euclidean plane
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, with V vertices (or corners), E edges and F faces (counting the exterior) Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
showed that V-E+F= 2. Thus 2 is called the Euler characteristic of the plane. By contrast, in 1813 Antoine-Jean Lhuilier showed that the Euler characteristic of the torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
is 0, since the complete graph
Complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
on seven points can be embedded into the torus. The Euler characteristic of other surfaces is a useful topological invariant, which has been extended to higher dimensions using Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s. In the mid nineteenth century, the Gauss–Bonnet theorem
Gauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
linked the Euler characteristic to the Gaussian curvature.
Lagrangian mechanics
Lagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
and Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
, when considered geometrically, are naturally manifold theories. All these use the notion of several characteristic axes or dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s (known as generalized coordinates
Generalized coordinates
In the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration....
in the latter two cases), but these dimensions do not lie along the physical dimensions of width, height, and breadth.
In the early 19th century the theory of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s succeeded in giving a basis for the theory of elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...
s, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
s of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen?
In the work of Niels Abel and Carl 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:...
, the answer was formulated: the resulting integral would involve functions of two complex variables, having four independent periods (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an abelian surface): what would now be called the Jacobian
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
of a hyperelliptic curve of genus 2.
Riemann
Bernhard RiemannBernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
was the first to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
term, Mannigfaltigkeit, which William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics...
translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s and 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 are named after Bernhard Riemann.
In 1857, Riemann introduced the concept 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 as part of a study of the process of analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
; Riemann surfaces are now recognized as one-dimensional complex manifolds. He also furthered the study of abelian and other multi-variable complex functions.
Contemporaries of Riemann
Johann Benedict ListingJohann Benedict Listing
Johann Benedict Listing was a German mathematician.J. B. Listing was born in Frankfurt and died in Göttingen. He first introduced the term "topology", in a famous article published in 1847, although he had used the term in correspondence some years earlier...
, inventor of the word "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...
", wrote an 1847 paper "Vorstudien zur Topologie" in which he defined a "complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
". He first defined the Möbius strip
Möbius strip
The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
in 1861 (rediscovered four years later by Möbius
August Ferdinand Möbius
August Ferdinand Möbius was a German mathematician and theoretical astronomer.He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict...
), as an example of a non-orientable surface.
After Abel, Jacobi, and Riemann, some of the most important contributors to the theory of abelian functions were Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
, Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
, Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
and Picard
Charles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...
. The subject was very popular at the time, already having a large literature. By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions.
Poincaré
Henri PoincaréHenri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
's 1895 paper Analysis Situs studied three-and-higher-dimensional manifolds, giving rigorous definitions of homology, homotopy (which had originally been defined in the context of late nineteenth-century knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, developed by Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
and others), and Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s and raised a question, today known as the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, based his new concept of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
. As of 2006, a consensus among experts is that recent work by Grigori Perelman may have answered this question, after nearly a century of effort by many mathematicians.
Later developments
Hermann WeylHermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
gave an intrinsic definition for differentiable manifolds in 1912. During the 1930s Hassler Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
and others clarified the foundational
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
theory.
The Whitney embedding theorem, which showed that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as in the extrinsic definition, showing that the two concepts of manifold were equivalent. Due to this unification, it is said to be the first complete exposition of the modern concept of manifold.
Eventually, in the 1920s
1920s
File:1920s decade montage.png|From left, clockwise: Third Tipperary Brigade Flying Column No. 2 under Sean Hogan during the Irish Civil War; Prohibition agents destroying barrels of alcohol in accordance to the 18th amendment, which made alcoholic beverages illegal throughout the entire decade; In...
, Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
laid the basis for the study of abelian functions in terms of complex tori. He also appears to have been the first to use the name "abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
"; in Romance languages
Romance languages
The Romance languages are a branch of the Indo-European language family, more precisely of the Italic languages subfamily, comprising all the languages that descend from Vulgar Latin, the language of ancient Rome...
, "variety" was used to translate Riemann's term "Mannigfaltigkeit". It was Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
in the 1940s
1940s
File:1940s decade montage.png|Above title bar: events which happened during World War II : From left to right: Troops in an LCVP landing craft approaching "Omaha" Beach on "D-Day"; Adolf Hitler visits Paris, soon after the Battle of France; The Holocaust occurred during the war as Nazi Germany...
who gave this subject its modern foundations in the language of algebraic geometry.
Sources
- Riemann, BernhardBernhard RiemannGeorg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse.- The 1851 doctoral thesis in which "manifold" (Mannigfaltigkeit) first appears.
- Riemann, Bernhard, On the Hypotheses which lie at the Bases of Geometry.
- The famous Göttingen inaugural lecture (Habilitationsschrift) of 1854.
- Early history of knot theory at St-Andrews history of mathematics website
- Early history of topology at St. Andrews
- H. Lange and Ch. Birkenhake, Complex Abelian Varieties, 1992, ISBN 0-387-54747-9
- A comprehensive treatment of the theory of abelian varieties, with an overview of the history the subject.
- André WeilAndré WeilAndré Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
: Courbes algébriques et variétés abéliennes, 1948- The first modern text on abelian varieties. In French.
- Henri PoincaréHenri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, Analysis Situs, Journal de l'École Polytechnique ser 2, 1 (1895) pages 1–123. - Henri Poincaré, Complément à l'Analysis Situs, Rendiconti del Circolo matematico di Palermo, 13 (1899) pages 285-343.
- Henri Poincaré, Second complément à l'Analysis Situs, Proceedings of the London Mathematical Society, 32 (1900), pages 277-308.
- Henri Poincaré, Sur certaines surfaces algébriques ; troisième complément à l'Analysis Situs, Bulletin de la Société mathématique de France, 30 (1902), pages 49–70.
- Henri Poincaré, Sur les cycles des surfaces algébriques ; quatrième complément à l'Analysis Situs, Journal de mathématiques pures et appliquées, 5° série, 8 (1902), pages 169-214.
- Henri Poincaré, Cinquième complément à l'analysis situs, Rendiconti del Circolo matematico di Palermo 18 (1904) pages 45–110.
- Erhard Scholz, Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincaré, Birkhäuser, 1980.
- A study of the genesis of the manifold concept. Based on the author's dissertation, directed by Egbert Brieskorn.