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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a manifold is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
that locally resembles
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
near each point. More precisely, an -dimensional manifold, or ''-manifold'' for short, is a topological space with the property that each point has a
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
that is
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to an open subset of -dimensional Euclidean space. One-dimensional manifolds include
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
s and
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
s, but not
figure eights
figure eights
. Two-dimensional manifolds are also called
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
s. Examples include the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
, the
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

sphere
, and the
torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ...

torus
, and also the
Klein bottle in three-dimensional space In topology, a branch of mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; it is a two-dimensional manifold against which a system for determining a normal vec ...

Klein bottle
and
real projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The concept of a manifold is central to many parts of
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
and modern
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of
systems of equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and as
graphs Graph may refer to: Mathematics *Graph (discrete mathematics) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (ge ...

graphs
of functions. The concept has applications in computer-graphics and augmented-reality given the need to associate pictures with coordinates (e.g.
CT scan A CT scan or computed tomography scan (formerly known as computed axial tomography or CAT scan) is a medical image, imaging Scientific technique, technique used in radiology to obtain detailed internal images of the body noninvasively for Diagno ...

CT scan
s). Manifolds can be equipped with additional structure. One important class of manifolds are
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
s; their
differentiable structureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
allows
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
to be done. A
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g'p'' on the tangent space ''T'p'M'' at each poin ...
on a manifold allows
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

distance
s and
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s to be measured.
Symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a Differentiable manifold#Definition, smooth manifold, M , equipped with a Closed and exact differential forms, closed nondegenerate form, nondegenerate Differential form ...
s serve as the
phase space In Dynamical systems theory, dynamical system theory, a phase space is a Space (mathematics), space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. ...

phase space
s in the
Hamiltonian formalism 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 ...
of classical mechanics, while four-dimensional
Lorentzian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a co ...
s model
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
in
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. The study of manifolds requires working knowledge of calculus and
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
.


Motivating examples


Circle

After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

unit circle
, ''x''2 + ''y''2 = 1, where the ''y''-coordinate is positive (indicated by the yellow arc in ''Figure 1''). Any point of this arc can be uniquely described by its ''x''-coordinate. So, projection onto the first coordinate is a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, and invertible,
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
from the upper arc to the
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
(−1, 1): \chi_(x,y) = x . \, Such functions along with the open regions they map are called ''
charts A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and c ...
''. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle: \begin \chi_\mathrm(x, y) &= x \\ \chi_\mathrm(x, y) &= y \\ \chi_\mathrm(x, y) &= y. \end Together, these parts cover the whole circle, and the four charts form an
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for the circle. The top and right charts, \chi_\mathrm and \chi_\mathrm respectively, overlap in their domain: their intersection lies in the quarter of the circle where both x and y-coordinates are positive. Each map this part into the interval (0,1), though differently. Thus a function T:(0,1)\rightarrow(0,1)=\chi_\mathrm \circ \chi^_\mathrm can be constructed, which takes values from the co-domain of \chi_\mathrm back to the circle using the inverse, followed by \chi_\mathrm back to the interval. For any number ''a'' in (0,1), then: \begin T(a) &= \chi_\mathrm\left(\chi_\mathrm^\left \rightright) \\ &= \chi_\mathrm\left(a, \sqrt\right) \\ &= \sqrt \end Such a function is called a ''transition map''. The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts \chi_\mathrm(x, y) = s = \frac and \chi_\mathrm(x, y) = t = \frac Here ''s'' is the slope of the line through the point at coordinates (''x'', ''y'') and the fixed pivot point (−1, 0); similarly, ''t'' is the opposite of the slope of the line through the points at coordinates (''x'', ''y'') and (+1, 0). The inverse mapping from ''s'' to (''x'', ''y'') is given by \begin x &= \frac \\
pt
pt
y &= \frac \end It can be confirmed that ''x''2 + ''y''2 = 1 for all values of ''s'' and ''t''. These two charts provide a second atlas for the circle, with the transition map t = \frac (that is, one has this relation between ''s'' and ''t'' for every point where ''s'' and ''t'' are both nonzero). Each chart omits a single point, either (−1, 0) for ''s'' or (+1, 0) for ''t'', so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.


Sphere

The
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

sphere
is an example of a surface. The
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
of
implicit equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: may be covered by an atlas of six
chart A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and ca ...
s: the plane divides the sphere into two half spheres ( and ), which may both be mapped on the disc by the projection on the plane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes. As for the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but the sphere cannot be covered by a single chart. This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

Earth
cannot have a plane representation consisting of a single
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

map
(also called "chart", see
nautical chart A nautical chart is a graphic representation of a sea area and adjacent coastal regions. Depending on the scale (map), scale of the chart, it may show depths of water and heights of land (topographic map), natural features of the seabed, details ...

nautical chart
), and therefore one needs
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atlas
es for covering the whole Earth surface.


Other curves

Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles. Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

parabola
, a
hyperbola In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it ...

hyperbola
, and the
locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * Locus (magazine), ''Locus'' (magazine), science fiction and fantasy magazine ...
of points on a
cubic curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(a closed loop piece and an open, infinite piece). However, excluded are examples like two touching circles that share a point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line segment, since deleting the center point from the "+" gives a space with four
components Component may refer to: In engineering, science, and technology Generic systems *System A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounde ...
(i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces; topological operations always preserve the number of pieces.


Mathematical definition

Informally, a manifold is a
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...
that is "modeled on" Euclidean space. There are many different kinds of manifolds. In
geometry and topology In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in Riemannian geometry#Local to glo ...
, all manifolds are
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological sp ...
s, possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance,
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
s have homeomorphisms on overlapping neighborhoods
diffeomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on the manifold as a whole. Formally, a (topological) manifold is a
second countable In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

Hausdorff space
that is locally homeomorphic to Euclidean space. ''Second countable'' and ''Hausdorff'' are point-set conditions; ''second countable'' excludes spaces which are in some sense 'too large' such as the
long lineLong line or longline may refer to: *''Long Line'', an album by Peter Wolf *Long line (topology), or Alexandroff line, a topological space *Long line (telecommunications), a transmission line in a long-distance communications network *Longline fishi ...
, while ''Hausdorff'' excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in non-Hausdorff manifolds). ''Locally homeomorphic'' to Euclidean space means that every point has a neighborhood
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to an open Euclidean ''n''-ball, \mathbf^n = \left\. More precisely, locally homeomorphic here means that each point ''m'' in the manifold ''M'' has an open neighborhood homeomorphic to an open neighborhood in Euclidean space. However, given such a homeomorphism, the pre-image of an \epsilon-ball gives a homeomorphism between the unit ball and a smaller neighborhood of ''m'', so this is no loss of generality. For topological or differentiable manifolds, one can also ask that every point have a neighborhood homeomorphic to all of Euclidean space (as this is diffeomorphic to the unit ball), but this cannot be done for
complex manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
s, as the complex unit ball is not
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
to complex space. Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed ''n''-ball), and such a space is called an ''n''-manifold; however, some authors admit manifolds where different points can have different
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
s. If a manifold has a fixed dimension, it is called a pure manifold. For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas the
disjoint union In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

disjoint union
of a sphere and a line in three-dimensional space is ''not'' a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is
locally constant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
), each connected component has a fixed dimension. Scheme-theoretically, a manifold is a
locally ringed spaceIn mathematics, a ringed space is a family of (Commutative ring, commutative) ring (mathematics), rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of Restriction (mathematics), restrictions. P ...
, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
.


Charts, atlases, and transition maps

The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold can be described using mathematical maps, called ''coordinate charts'', collected in a mathematical ''atlas''. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure.


Charts

A ''coordinate map'', a ''coordinate chart'', or simply a ''chart'', of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is a subset of some Euclidean space \R^n and interest focuses on the topological structure. This structure is preserved by
homeomorphisms and a donut (torus In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerne ...
, invertible maps that are continuous in both directions. In the case of a differentiable manifold, a set of ''charts'' called an ''atlas'' allows us to do calculus on manifolds.
Polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Polar coordinates
, for example, form a chart for the plane \R^2 minus the positive ''x''-axis and the origin. Another example of a chart is the map χtop mentioned above, a chart for the circle.


Atlases

The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an ''
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''. An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union is also an atlas. The atlas containing all possible charts consistent with a given atlas is called the ''maximal atlas'' (i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations).


Transition maps

Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Asia may both contain Moscow. Given two overlapping charts, a ''transition function'' can be defined which goes from an open ball in \R^n to the manifold and then back to another (or perhaps the same) open ball in \R^n. The resultant map, like the map ''T'' in the circle example above, is called a ''change of coordinates'', a ''coordinate transformation'', a ''transition function'', or a ''transition map''.


Additional structure

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure, the structure transfers to the manifold. This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of \R^n (that is, if they are
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s), the differential structure transfers to the manifold and turns it into a differentiable manifold.
Complex manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
s are introduced in an analogous way by requiring that the transition functions of an atlas are
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
s. For
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a Differentiable manifold#Definition, smooth manifold, M , equipped with a Closed and exact differential forms, closed nondegenerate form, nondegenerate Differential form ...
s, the transition functions must be
symplectomorphismIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s. The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called ''compatible''. These notions are made precise in general through the use of
pseudogroupIn mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group (mathematics), group, originating however from the geometric app ...
s.


Manifold with boundary

A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a
2-manifold with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the Manifold#Manifold_with_boundary, boundaries of three-dimensional solids; for e ...
with a 1-dimensional boundary. The boundary of an ''n''-manifold with boundary is an -manifold. A
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
(circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

square
with interior is also a 2-manifold with boundary. A
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
(sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (See also
Boundary (topology) In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
). In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open ''n''-ball Every boundary point has a neighborhood homeomorphic to the "half" ''n''-ball . The homeomorphism must send each boundary point to a point with ''x''1 = 0.


Boundary and interior

Let ''M'' be a manifold with boundary. The interior of ''M'', denoted Int ''M'', is the set of points in ''M'' which have neighborhoods homeomorphic to an open subset of \R^n. The boundary of ''M'', denoted ∂''M'', is the
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
of Int''M'' in ''M''. The boundary points can be characterized as those points which land on the boundary hyperplane of \R^n_+ under some coordinate chart. If ''M'' is a manifold with boundary of dimension ''n'', then Int''M'' is a manifold (without boundary) of dimension ''n'' and ∂''M'' is a manifold (without boundary) of dimension .


Construction

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint.


Charts

Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of \R^2 is identified, and then an atlas covering this subset is constructed. The concept of ''manifold'' grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:


Sphere with charts

A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of \R^3: S = \left\. The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of \R^3. Consider the northern hemisphere, which is the part with positive ''z'' coordinate (coloured red in the picture on the right). The function χ defined by \chi(x, y, z) = (x, y),\ maps the northern hemisphere to the open
unit disc An open Euclidean unit disk In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemati ...

unit disc
by projecting it on the (''x'', ''y'') plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (''x'', ''z'') plane and two charts projecting on the (''y'', ''z'') plane, an atlas of six charts is obtained which covers the entire sphere. This can be easily generalized to higher-dimensional spheres.


Patchwork

A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold. The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold. This can be illustrated with the transition map ''t'' = 1''s'' from the second half of the circle example. Start with two copies of the line. Use the coordinate ''s'' for the first copy, and ''t'' for the second copy. Now, glue both copies together by identifying the point ''t'' on the second copy with the point ''s'' = 1''t'' on the first copy (the points ''t'' = 0 and ''s'' = 0 are not identified with any point on the first and second copy, respectively). This gives a circle.


Intrinsic and extrinsic view

The first construction and this construction are very similar, but represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. This is the ''extrinsic view''. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space, it is always clear whether a vector at some point is
tangential In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely cl ...
or
normal
normal
to some surface through that point. The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the ''intrinsic view''. It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle.


''n''-Sphere as a patchwork

The
''n''-sphere
''n''-sphere
S''n'' is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An ''n''-sphere S''n'' can be constructed by gluing together two copies of \R^n. The transition map between them is
inversion in a sphere In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
, defined as \R^n \setminus \ \to \R^n \setminus \: x \mapsto x/\, x\, ^2. This function is its own inverse and thus can be used in both directions. As the transition map is a
smooth function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...

smooth function
, this atlas defines a smooth manifold. In the case ''n'' = 1, the example simplifies to the circle example given earlier.


Identifying points of a manifold

It is possible to define different points of a manifold to be same. This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds,
orbifold In the mathematical disciplines of topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which ...
s and
CW complex A CW complex is a kind of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantiti ...
es are considered to be relatively
well-behaved In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is ...
. An example of a quotient space of a manifold that is also a manifold is the
real projective spaceIn mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n' ...
, identified as a quotient space of the corresponding sphere. One method of identifying points (gluing them together) is through a right (or left) action of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, which
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum), often referred to simply as Acts, or formally the Book of Acts, is the fifth book of the New Testament The New Te ...
on the manifold. Two points are identified if one is moved onto the other by some group element. If ''M'' is the manifold and ''G'' is the group, the resulting quotient space is denoted by ''M'' / ''G'' (or ''G'' \ ''M''). Manifolds which can be constructed by identifying points include
tori Tori may refer to: Places * Tori (Georgia), an historic province of the nation of Georgia * Tori Parish, Pärnu County, Estonia ** Tori, Estonia * Tori, Ghana, a village in the kingdom of Chumburung, Ghana * Tori, Järva County, Estonia * Tori, ...

tori
and
real projective spaceIn mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n' ...
s (starting with a plane and a sphere, respectively).


Gluing along boundaries

Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together. Formally, the gluing is defined by a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
between the two boundaries. Two points are identified when they are mapped onto each other. For a topological manifold, this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly, for a differentiable manifold, it has to be a diffeomorphism. For other manifolds, other structures should be preserved. A finite cylinder may be constructed as a manifold by starting with a strip ,1nbsp;×  ,1and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A
projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
may be obtained by gluing a sphere with a hole in it to a
Möbius strip In mathematics, a Möbius strip, band, or loop ( , ; ), also spelled ''Mobius'' or ''Moebius'', is a Surface (topology), surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary (topology), boundary ...

Möbius strip
along their respective circular boundaries.


Cartesian products

The
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of manifolds is also a manifold. The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

cylinder
s, for example, as S1 × S1 and S1 ×  ,1 respectively.


History

The study of manifolds combines many important areas of 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 (geo ...

curve
s and surfaces 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 matrix (mat ...
and topology.


Early development

Before the modern concept of a manifold there were several important results.
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 geome ...
considers spaces where
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
's
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

parallel postulate
fails.
Saccheri Image:Saccheri 1733 - Euclide Ab Omni Naevo Vindicatus.gif, upThe Book frontispiece, frontispiece of "Euclides ab omni nævo vindicatus" (1733). Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italy, Italian Jesuit pries ...
first studied such geometries in 1733, but sought only to disprove them. Carl Friedrich Gauss, Gauss, János Bolyai, Bolyai and Nikolai Ivanovich Lobachevsky, Lobachevsky independently discovered them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian 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 File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic and extrinsic properties, 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 topological property, property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with ''V'' vertex (geometry), vertices (or corners), ''E'' edges, and ''F'' faces,V - E + F = 2.\ The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a topological map with ''V'' vertices, ''E'' edges, and ''F'' faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope. Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a
torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ...

torus
can be sliced open by its 'parallel' and 'meridian' circles, creating a map with ''V'' = 1 vertex, ''E'' = 2 edges, and ''F'' = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.


Synthesis

Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Abelian variety, Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables. Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Siméon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought to be points of an abstract space,
phase space In Dynamical systems theory, dynamical system theory, a phase space is a Space (mathematics), space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. ...

phase space
in Lagrangian mechanics, Lagrangian and Hamiltonian mechanics, Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph-Louis Lagrange, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology. Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name ''manifold'' comes from Riemann's original German language, 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 Mathematical induction, 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 manifolds and Riemann surfaces are named after Riemann.


Poincaré's definition

In his very influential paper, Analysis Situs (paper), Analysis Situs, Henri Poincaré gave a definition of a differentiable manifold (''variété'') which served as a precursor to the modern concept of a manifold. In the first section of Analysis Situs, Poincaré defines a manifold as the level set of a continuously differentiable function between Euclidean spaces that satisfies the nondegeneracy hypothesis of the implicit function theorem. In the third section, he begins by remarking that the graph of a function, graph of a continuously differentiable function is a manifold in the latter sense. He then proposes a new, more general, definition of manifold based on a 'chain of manifolds' (''une chaîne des variétés''). Poincaré's notion of a ''chain of manifolds'' is a precursor to the modern notion of atlas. In particular, he considers two manifolds defined respectively as graphs of functions \theta(y) and \theta'\left(y'\right) . If these manifolds overlap (''a une partie commune''), then he requires that the coordinates y depend continuously differentiably on the coordinates y' and vice versa (...les y sont fonctions analytiques des y' et inversement'''). In this way he introduces a precursor to the notion of a #Charts, chart and of a #Transition maps, transition map. For example, the unit circle in the plane can be thought of as the graph of the function y = \sqrt or else the function y = -\sqrt in a neighborhood of every point except the points (1, 0) and (−1, 0); and in a neighborhood of those points, it can be thought of as the graph of, respectively, x = \sqrt and x = -\sqrt. The circle can be represented by a graph in the neighborhood of every point because the left hand side of its defining equation x^2 + y^2 - 1 = 0 has nonzero gradient at every point of the circle. By the implicit function theorem, every submanifold of Euclidean space is locally the graph of a function. Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
that followed shortly. During the 1930s Hassler Whitney and others clarified the foundations of mathematics, 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. Notably, the Whitney embedding theorem showed that the intrinsic definition in terms of charts was equivalent to Poincaré's definition in terms of subsets of Euclidean space.


Topology of manifolds: highlights

Two-dimensional manifolds, also known as a 2D ''surfaces'' embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century, Grigori Perelman proved the Poincaré conjecture (see the Poincaré conjecture#Solution, Solution of the Poincaré conjecture). William Thurston's geometrization conjecture, geometrization program, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recent progress in theoretical physics (Yang–Mills theory), where they serve as a substitute for ordinary 'flat'
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
. Andrey Markov (Soviet mathematician), Andrey Markov Jr. showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higher-dimensional manifolds, including Generalized Poincaré conjecture, analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov (mathematician), Sergei Novikov. A very pervasive and flexible technique underlying much work on the differential topology, topology of manifolds is Morse theory.


Additional structure


Topological manifolds

The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space \R^n. By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space Local homeomorphism, locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijection, bijective continuous function (topology), continuous function whose inverse is also continuous) mapping that neighbourhood to \R^n. These homeomorphisms are the charts of the manifold. A ''topological'' manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any ''particular'' and ''consistent'' choice of such concepts. In order to discuss such properties for a manifold, one needs to specify further structure and consider #Differentiable manifolds, differentiable manifolds and #Riemannian manifolds, Riemannian manifolds discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles. Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff space, Hausdorff and second-countable space, second countable. The ''dimension'' of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number ''n'' in the definition). All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension.


Differentiable manifolds

For most applications, a special kind of topological manifold, namely, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
on a differentiable manifold. Each point of an ''n''-dimensional differentiable manifold has a tangent space. This is an ''n''-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is, infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic function, analytic (they can be expressed as power series). The sphere can be given analytic structure, as can most familiar curves and surfaces. A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.


Riemannian manifolds

To measure distances and angles on manifolds, the manifold must be Riemannian. A ''Riemannian manifold'' is a differentiable manifold in which each tangent space is equipped with an Inner product space, inner product in a manner which varies smoothly from point to point. Given two tangent vectors and , the inner product gives a real number. The dot product, dot (or scalar) product is a typical example of an inner product. This allows one to define various notions such as length,
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s, areas (or volumes), curvature and divergence of vector fields. All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all -spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it.


Finsler manifolds

A Finsler manifold allows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with a normed space, norm, , , ·, , , in a manner which varies smoothly from point to point. This norm can be extended to a Metric (mathematics), metric, defining the length of a curve; but it cannot in general be used to define an inner product. Any Riemannian manifold is a Finsler manifold.


Lie groups

Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
which is such that the group operations are defined by smooth maps. A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact space, compact Lie group is the circle: the group operation is simply rotation. This group, known as U(1), can be also characterised as the group of complex numbers of absolute value, modulus 1 with multiplication as the group operation. Other examples of Lie groups include special groups of Matrix (mathematics), matrices, which are all subgroups of the general linear group, the group of ''n'' by ''n'' matrices with non-zero determinant. If the matrix entries are real numbers, this will be an ''n''2-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are ''n''(''n''−1)/2 dimensional manifolds, where ''n''−1 is the dimension of the sphere. Further examples can be found in the table of Lie groups.


Other types of manifolds

* A ''
complex manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
'' is a manifold whose charts take values in \Complex^n and whose transition functions are
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
on the overlaps. These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. An ''n''-dimensional complex manifold has dimension 2''n'' as a real differentiable manifold. * A ''CR manifold'' is a manifold modeled on boundaries of domains in \Complex^n. * 'Infinite dimensional manifolds': to allow for infinite dimensions, one may consider Banach manifolds which are locally homeomorphic to Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic to Fréchet spaces. * A ''
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a Differentiable manifold#Definition, smooth manifold, M , equipped with a Closed and exact differential forms, closed nondegenerate form, nondegenerate Differential form ...
'' is a kind of manifold which is used to represent the phase spaces in classical mechanics. They are endowed with a differential form, 2-form that defines the Poisson bracket. A closely related type of manifold is a Contact geometry, contact manifold. * A ''combinatorial manifold'' is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes. * A ''digital manifold'' is a special kind of combinatorial manifold which is defined in digital space. See digital topology


Classification and invariants

Different notions of manifolds have different notions of classification and invariant; in this section we focus on smooth closed manifolds. The classification of smooth closed manifolds is well understood ''in principle'', except in 4-manifold, dimension 4: in low dimensions (2 and 3) it is geometric, via the uniformization theorem and the solution of the Poincaré conjecture, and in high dimension (5 and above) it is algebraic, via surgery theory. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions. Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a complete set of invariants. This is much harder in higher dimensions: higher-dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object. However, one can determine if two manifolds are ''different'' if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as invariant (mathematics), invariants, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are ''invariant'' under different descriptions. Naively, one could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. Unfortunately, it is known that for manifolds of dimension 4 and higher, classification of manifolds#Computability, no program exists that can decide whether two manifolds are diffeomorphic. Smooth manifolds have Classification of manifolds#Enumeration versus invariants, a rich set of invariants, coming from point-set topology, classic algebraic topology, and geometric topology. The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant, also detected by singular homology, homology) and genus (mathematics), genus (a homological invariant). Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of Riemannian manifolds, curvature of a Riemannian manifold and the torsion (differential geometry), torsion of a manifold equipped with an affine connection. This distinction between local invariants and no local invariants is a common way to distinguish between Geometry and topology#Local versus global structure, geometry and topology. All invariants of a smooth closed manifold are thus global. Algebraic topology is a source of a number of important global invariant properties. Some key criteria include the ''simply connected'' property and orientability (see below). Indeed, several branches of mathematics, such as homology (mathematics), homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds.


Surfaces


Orientability

In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation. Consider a topological manifold with charts mapping to \R^n. Given an basis (linear algebra)#Ordered bases and coordinates, ordered basis for \R^n, a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are ''Orientability, orientable'' manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable. Some illustrative examples of non-orientable manifolds include: (1) the
Möbius strip In mathematics, a Möbius strip, band, or loop ( , ; ), also spelled ''Mobius'' or ''Moebius'', is a Surface (topology), surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary (topology), boundary ...

Möbius strip
, which is a manifold with boundary, (2) the
Klein bottle in three-dimensional space In topology, a branch of mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; it is a two-dimensional manifold against which a system for determining a normal vec ...

Klein bottle
, which must intersect itself in its 3-space representation, and (3) the
real projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which arises naturally in geometry.


Möbius strip

Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a ''single'' side. Similarly to the Klein Bottle below, this two dimensional surface would need to intersect itself in two dimensions, but can easily be constructed in three or more dimensions.


Klein bottle

Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. In three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space.


Real projective plane

Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called ''antipodes''. Although there is no way to do so physically, it is possible (by considering a quotient space) to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin project to the same "point" on this "plane".


Genus and the Euler characteristic

For two dimensional manifolds a key invariant property is the genus (mathematics), genus, or "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed, it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of Euler characteristic, and more generally Betti numbers and homology (mathematics), homology and cohomology.


Maps of manifolds

Just as there are various types of manifolds, there are various types of maps of manifolds. In addition to continuous functions and smooth functions generally, there are maps with special properties. In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersion (mathematics), immersions, Submersion (mathematics), submersions, covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem. In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem.


Scalar-valued functions

A basic example of maps between manifolds are scalar-valued functions on a manifold, f\colon M \to \R or f\colon M \to \Complex, sometimes called regular functions or functional (mathematics), functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold. In geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, while in mathematical analysis, one often studies solution to partial differential equations, an important example of which is harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to such functions as the spherical harmonics, and to heat kernel methods of studying manifolds, such as hearing the shape of a drum and some proofs of the Atiyah–Singer index theorem.


Generalizations of manifolds

;Infinite dimensional manifolds: The definition of a manifold can be generalized by dropping the requirement of finite dimensionality. Thus an infinite dimensional manifold is a topological space locally homeomorphic to a topological vector space over the reals. This omits the point-set axioms, allowing higher cardinalities and non-Hausdorff manifolds; and it omits finite dimension, allowing structures such as Hilbert manifolds to be modeled on Hilbert spaces, Banach manifolds to be modeled on Banach spaces, and Fréchet manifolds to be modeled on Fréchet spaces. Usually one relaxes one or the other condition: manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis. ;Orbifolds: An
orbifold In the mathematical disciplines of topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which ...
is a generalization of manifold allowing for certain kinds of "mathematical singularity, singularities" in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (''e.g.'' Euclidean space) by the Group action (mathematics), actions of various finite groups. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense. ;Algebraic varieties and schemes: Non-singular algebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subsets of Euclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. Scheme (mathematics), Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically using sheaf (mathematics), sheaves instead of atlases. :Because of Mathematical singularity, singular points, a variety is in general not a manifold, though linguistically the French ''variété'', German ''Mannigfaltigkeit'' and English ''manifold'' are largely synonymous. In French an algebraic variety is called ''une :fr:variété algébrique, variété algébrique'' (an ''algebraic variety''), while a smooth manifold is called ''une :fr:variété différentielle, variété différentielle'' (a ''differential variety''). ;Stratified space: A "stratified space" is a space that can be divided into pieces ("strata"), with each stratum a manifold, with the strata fitting together in prescribed ways (formally, a Filtration (mathematics), filtration by closed subsets). There are various technical definitions, notably a Whitney stratified space (see Whitney conditions) for smooth manifolds and a topologically stratified space for topological manifolds. Basic examples include manifold with boundary (top dimensional manifold and codimension 1 boundary) and manifolds with corners (top dimensional manifold, codimension 1 boundary, codimension 2 corners). Whitney stratified spaces are a broad class of spaces, including algebraic varieties, analytic varieties, semialgebraic sets, and subanalytic sets. ;CW-complexes: A
CW complex A CW complex is a kind of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantiti ...
is a topological space formed by gluing disks of different dimensionality together. In general the resulting space is singular, hence not a manifold. However, they are of central interest in algebraic topology, especially in homotopy theory. ;Homology manifolds: A homology manifold is a space that behaves like a manifold from the point of view of homology theory. These are not all manifolds, but (in high dimension) can be analyzed by surgery theory similarly to manifolds, and failure to be a manifold is a local obstruction, as in surgery theory. ;Differential spaces: Let M be a nonempty set. Suppose that some family of real functions on M was chosen. Denote it by C \subseteq \R^M. It is an algebra with respect to the pointwise addition and multiplication. Let M be equipped with the topology induced by C. Suppose also that the following conditions hold. First: for every H \in C^\infty\left(\R^i\right), where i \in \N, and arbitrary f_1, \dots , f_n \in C, the composition H \circ \left(f_1, \dots, f_n\right) \in C. Second: every function, which in every point of M locally coincides with some function from C, also belongs to C. A pair (M, C) for which the above conditions hold, is called a Sikorski differential space.


See also

* * : statistics on manifolds * * *


By dimension

* * * *


Notes


References

* Michael Freedman, Freedman, Michael H., and Quinn, Frank (1990) ''Topology of 4-Manifolds''. Princeton University Press. . * Victor Guillemin, Guillemin, Victor and Pollack, Alan (1974) ''Differential Topology''. Prentice-Hall. . Advanced undergraduate / first-year graduate text inspired by Milnor. * Hempel, John (1976) ''3-Manifolds''. Princeton University Press. . * Morris Hirsch, Hirsch, Morris, (1997) ''Differential Topology''. Springer Verlag. . The most complete account, with historical insights and excellent, but difficult, problems. The standard reference for those wishing to have a deep understanding of the subject. * Robion Kirby, Kirby, Robion C. and Siebenmann, Laurence C. (1977) ''Foundational Essays on Topological Manifolds. Smoothings, and Triangulations''. Princeton University Press. . A detailed study of the category theory, category of topological manifolds. * Lee, John M. (2000) ''Introduction to Topological Manifolds''. Springer-Verlag. . Detailed and comprehensive first-year graduate text. * Lee, John M. (2003)
Introduction to Smooth Manifolds
'. Springer-Verlag. . Detailed and comprehensive first-year graduate text; sequel to ''Introduction to Topological Manifolds''. * William S. Massey, Massey, William S. (1977) ''Algebraic Topology: An Introduction''. Springer-Verlag. . * John Milnor, Milnor, John (1997) ''Topology from the Differentiable Viewpoint''. Princeton University Press. . Classic brief introduction to differential topology. * James Munkres, Munkres, James R. (1991)
Analysis on Manifolds
'. Addison-Wesley (reprinted by Westview Press) . Undergraduate text treating manifolds in \R^n. * James Munkres, Munkres, James R. (2000) ''Topology''. Prentice Hall. . * Neuwirth, L. P., ed. (1975) ''Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox''. Princeton University Press. . * Bernhard Riemann, Riemann, Bernhard, ''Gesammelte mathematische Werke und wissenschaftlicher Nachlass'', Sändig Reprint. . **
Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse.
' The 1851 doctoral thesis in which "manifold" (''Mannigfaltigkeit'') first appears. **
Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.
' The 1854 Göttingen inaugural lecture (''Habilitationsschrift''). * Michael Spivak, Spivak, Michael (1965)
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
'. W.A. Benjamin Inc. (reprinted by Addison-Wesley and Westview Press). . Calculus on Manifolds (book), Famously terse advanced undergraduate / first-year graduate text. * Michael Spivak, Spivak, Michael (1999) ''A Comprehensive Introduction to Differential Geometry'' (3rd edition) Publish or Perish Inc. Encyclopedic five-volume series presenting a systematic treatment of the theory of manifolds, Riemannian geometry, classical differential geometry, and numerous other topics at the first- and second-year graduate levels. * . Concise first-year graduate text.


External links

*
Dimensions-math.org
(A film explaining and visualizing manifolds up to fourth dimension.) * Th
manifold atlas
project of th
Max Planck Institute for Mathematics in Bonn
{{Authority control Manifolds, Geometry processing