The study of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s combines many important areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
: it generalizes concepts such as
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s and
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
s as well as ideas from
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
and
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. Certain special classes of manifolds also have additional algebraic structure; they may behave like
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
s, for instance. In that case, they are called
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s. Alternatively, they may be described by
polynomial equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
, in which case they are called
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
, and if they additionally carry a group structure, they are called
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
s.
Nomenclature
The term "manifold" comes from German ''Mannigfaltigkeit,'' by
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
.
In English, "
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
" refers to spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
.
In Romance languages, manifold is translated as "variety" – such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the French word "
variété topologique" means
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
. In the same vein, the Japanese word "" (tayōtai) also encompasses both manifold and variety. ("" (tayō) means various.)
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 geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, which considers spaces where
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's
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. It states that, in two-dimensional geometry:
''If a line segment ...
fails.
Saccheri
Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician.
Saccheri was born in Sanremo. He entered the Jesuit order in 1685 and was ordained as a priest in 1694. ...
first studied this geometry in 1733.
Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...
,
Bolyai, and
Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
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 is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
; these are called
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
and
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
. In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positive
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
, respectively.
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
may have been the first to consider abstract spaces as mathematical objects in their own right. His
theorema egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
gives a method for computing the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
without considering the
ambient space
An ambient space or ambient configuration space is the space surrounding an object.
While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former perceives space as ''navigable'', while the latt ...
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
example of an intrinsic
property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
of a manifold is the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
. For a non-intersecting
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, with ''V'' vertices (or corners), ''E'' edges and ''F'' faces (counting the exterior)
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
is 0, since the
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. A complete digraph is a directed graph in which every pair of distinct vertices is c ...
on seven points can be embedded into the torus. The Euler characteristic of other surfaces is a useful
topological invariant
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces ...
, which has been extended to higher
dimensions
In physics and mathematics, the dimension of a mathematical 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 (1D) because only one coordina ...
using
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s. In the mid nineteenth century, the
Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.
In the simplest application, the case of a ...
linked the Euler characteristic to the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
.
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
and
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, when considered geometrically, are naturally manifold theories. All these use the notion of several characteristic
axes
Axes, plural of ''axe'' and of ''axis'', may refer to
* ''Axes'' (album), a 2005 rock album by the British band Electrelane
* a possibly still empty plot (graphics)
See also
*Axess (disambiguation)
*Axxess (disambiguation) Axxess may refer to:
...
or
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s (known as
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
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
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
s succeeded in giving a basis for the theory of
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
s of
cubic and
quartic polynomial
In algebra, a quartic function is a function (mathematics), function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
where ''a'' is nonzero,
which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial.
A ''qua ...
s. When those were replaced by polynomials of higher degree, say
quintics, what would happen?
In the work of
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
and
Carl Jacobi, the answer was formulated: the
resulting integral would involve functions of
two complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
, 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
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...
of genus 2''.
Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
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(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
term, ''Mannigfaltigkeit'', which
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) 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 ...
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
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
, 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
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s and
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s are named after Bernhard Riemann.
In 1857, Riemann introduced the concept of
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s as part of a study of the process of
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
; 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 Listing
Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician.
J. B. Listing was born in Frankfurt and died in Göttingen. He first introduced the term "topology" to replace the older term "geometria situs" (also called ...
, inventor of the word "
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
", wrote an 1847 paper "Vorstudien zur Topologie" in which he defined a "
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
". He first defined the
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
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 function
In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 func ...
s were
Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
,
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.
Poincaré
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
's 1895 paper
Analysis Situs studied three-and-higher-dimensional manifolds (which he called "varieties"), giving rigorous definitions of homology, homotopy, and
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s and raised a question, today known as the
Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
, based his new concept of the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
. In 2003,
Grigori Perelman proved the conjecture using
Richard S. Hamilton's
Ricci flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analo ...
, this is after nearly a century of effort by many mathematicians.
Later developments
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
gave an intrinsic definition for differentiable manifolds in 1912. During the 1930s
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
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
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
theory.
The
Whitney embedding theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
*The strong Whitney embedding theorem states that any differentiable manifold, smooth real numbers, real -dimension (math ...
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
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
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 mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 func ...
"; in
Romance languages
The Romance languages, sometimes referred to as Latin languages or Neo-Latin languages, are the various modern languages that evolved from Vulgar Latin. They are the only extant subgroup of the Italic languages in the Indo-European language fam ...
, "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.
Sources
*
Riemann, Bernhard''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.
*
ttp://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Topology_in_mathematics.html Early history of topology at St. Andrews* H. Lange and Ch. Birkenhake, Complex Abelian Varieties, 1992,
** A comprehensive treatment of the theory of abelian varieties, with an overview of the history the subject.
*
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
: Courbes algébriques et variétés abéliennes, 1948
** The first modern text on abelian varieties. In French.
*
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, ''Analysis Situs'', Journal de l'École Polytechnique ser 2, 1 (1895) pages 1–123.
* Henri Poincaré, ''Complément à l'Analysis Situs'',
, 13 (1899) pages 285–343.
* Henri Poincaré, ''Second complément à l'Analysis Situs'',
Proceedings of the London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
, 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
Erhard Scholz (born 1947) is a German historian of mathematics with interests in the history of mathematics in the 19th and 20th centuries, historical perspective on the philosophy of mathematics and science, and Hermann Weyl's geometrical methods ...
, ''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.
{{DEFAULTSORT:History Of Manifolds And Varieties
Manifolds and varieties
Manifolds