On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations A sphere is not a Euclidean space, but locally the laws of Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. A sphere can be represented by a collection of two dimensional maps, therefore a sphere is a manifold.

A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In discussing manifolds, the idea of dimension is important. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it For example, lines are one-dimensional, and planes two-dimensional.

In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In a two-manifold, every point has a neighborhood that looks like a disk. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. Examples include a plane, the surface of a sphere, and the surface of a torus. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

A precise mathematical definition of a manifold is given below. To fully understand the mathematics behind manifolds, it is necessary to know elementary concepts regarding sets and functions, and helpful to have a working knowledge of calculus and topology. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

Motivational examples

Circle

Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.

The circle is the simplest example of a topological manifold after a line. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Topology ignores bending, so a small piece of a circle is exactly the same as a small piece of a line. Consider, for instance, the top half of the unit circle, x² + y² = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). In Mathematics, a unit circle is In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Any point of this semicircle can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the upper semicircle to the open interval (−1,1):

Such functions along with the open regions they map are called charts. In Mathematics, a projection is any one of several different types of functions mappings operations or transformations for example the following In In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle. Together, these parts cover the whole circle and the four charts form an atlas for the circle. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how

The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χ_top and χ_right each map this part into the interval (0,1). Thus a function T from (0,1) to itself can be constructed, which first uses the inverse of the top chart to reach the circle and then follows the right chart back to the interval. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B Let a be any number in (0,1), then:

Such a function is called a transition map.

Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.

The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts

and

Here s is the slope of the line through the point at coordinates (x,y) and the fixed pivot point (−1,0); t is the mirror image, with pivot point (+1,0). The inverse mapping from s to (x,y) is given by

It can easily be confirmed that x²+y² = 1 for all values of the slope s. These two charts provide a second atlas for the circle, with

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. Topology can prove 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 "glueing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.

Other curves

Four manifolds from algebraic curves: ■ circles, ■ parabola, ■ hyperbola, ■ cubic.

Manifolds need not be connected (all in "one piece"); thus a pair of separate circles is also a manifold. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of They need not be closed; thus a line segment without its end points is a manifold. See also Classification of manifolds#Point-set In Mathematics, a closed manifold is type of Topological space, namely a compact And they need not be finite; thus a parabola is a manifold. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular Putting these freedoms together, two other example manifolds are a hyperbola (two open, infinite pieces) and the locus of points on the cubic curve y² = x³−x (a closed loop piece and an open, infinite piece). In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property In Mathematics, a cubic plane curve is a Plane algebraic curve C defined by a cubic equation F ( x, y,

However, we exclude examples like two touching circles that share a point to form a figure-8; at the shared point we cannot create a satisfactory chart. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line.

Enriched circle

Viewed using calculus, the circle transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The transition map T, and all the others, are differentiable on (0, 1); therefore, with this atlas the circle is a differentiable manifold. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. It is also smooth and analytic because the transition functions have these properties as well.

Other circle properties allow it to meet the requirements of more specialized types of manifold. For example, the circle has a notion of distance between two points, the arc-length between the points; hence it is a Riemannian manifold. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M

History

For more details on this topic, see History of manifolds and varieties. The study of Manifolds combines many important areas of Mathematics: it generalizes concepts such as Curves and Surfaces as well as ideas from Linear

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Linear algebra is the branch of Mathematics concerned with Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

Prehistory

Before the modern concept of a manifold there were several important results.

Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Saccheri first studied them in 1733. Giovanni Girolamo Saccheri ( September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician Year 1733 ( MDCCXXXIII) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Lobachevsky, Bolyai, and Riemann developed them 100 years later. Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean 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 Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature, respectively. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry

Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Gauss's Theorema Egregium (Latin "Remarkable Theorem" is a foundational result in Differential geometry proved by Carl Friedrich Gauss that concerns the In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object. 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 property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space.

Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges, and F faces,

V-E+F= 2. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations

The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a '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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe ^[1] Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a 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. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar 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 Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature. The Gauss–Bonnet theorem or Gauss–Bonnet formula in Differential geometry is an important statement about Surfaces which connects their geometry (in

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 Jacobians. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation Carl Gustav Jacob Jacobi ( December 10, 1804 - February 18, 1851) was a Prussian Mathematician, widely considered to be In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables, although these ideas were way ahead of their time. In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function.

Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Simeon Poisson, Jacobi, and William Rowan Hamilton. Analytical mechanics is a term used for a refined highly mathematical form of Classical mechanics, constructed from the Eighteenth century onwards as a formulation Siméon-Denis Poisson (21 June 1781 &ndash 25 April 1840 was a French Mathematician, Geometer, and Physicist. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. 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. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately 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. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph 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. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

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 term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness". The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages. William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Bernhard Riemann. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional

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 that followed shortly. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. Hassler Whitney ( 23 March 1907 &ndash 10 May 1989) was an American Mathematician. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Topology of manifolds: highlights

Two-dimensional manifolds, also known as surfaces, 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. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional Poul Heegaard ( November 2, 1871 - February 7, 1948) was a Danish Mathematician active in the field of Topology Max Dehn ( November 13, 1878, Hamburg, Germany – June 27, 1952, Black Mountain, North Carolina, Henri Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among After nearly a century of effort by many mathematicians, starting with Poincaré himself, a consensus among experts (as of 2006) is that Grigori Perelman has proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). Grigori Yakovlevich Perelman (Григорий Яковлевич Перельман born 13 June 1966 in Leningrad, USSR (now St This entry describes the solution of the Poincaré conjecture at a level intended for the general public Bill Thurston's geometrization program, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. William Paul Thurston (born October 30, 1946) is an American Mathematician. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures 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' space-time. Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, U Simon Kirwan Donaldson (born August 20 1957 in Cambridge, England) is an English mathematician famous for his work on the Topology of Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Important work on higher-dimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov. René Thom ( September 2, 1923 – October 25, 2002) was a French Mathematician. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential Stephen Smale (born July 15, 1930) is an American Mathematician from Flint Michigan. Sergei Petrovich Novikov (also Serguei) ( Russian Сергей Петрович Новиков (born 20 March 1938) is a Russian One of the most pervasive and flexible techniques underlying much work on the topology of manifolds is Morse theory. In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential "Morse function" redirects here In another context a "Morse function" can also mean an Anharmonic oscillator.

Mathematical definition

For more details on this topic, see Categories of manifolds. In Mathematics, specifically Geometry and topology, there are many different notions of Manifold, with more or less structure and corresponding notions of"map

Informally, a manifold is a space that is "modeled on" Euclidean space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

There are many different kinds of manifolds and generalizations. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, most often a differentiable structure. In Mathematics, geometry and topology is an Umbrella term for Geometry and Topology, as the line between these two is often blurred most In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In terms of constructing manifolds via patching, a manifold has an additional structure if the transition maps between different patches satisfy axioms beyond just continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, and so differentiable on the manifold as a whole. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

Formally, a topological manifold^[2] is a second countable Hausdorff space that is locally homeomorphic to Euclidean space. In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability " In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology, a local homeomorphism is a map from one Topological space to another that respects locally the topological structure of the two spaces

Second countable and Hausdorff are point-set conditions; second countable excludes spaces of higher cardinality such as the long line, while Hausdorff excludes spaces such as "the line with two origins" (these generalized manifolds are discussed in non-Hausdorff manifolds). In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Topology, the long line (or Alexandroff line) is a Topological space analogous to the Real line, but much longer In Mathematics, it is a usual axiom of a Manifold to be a Hausdorff space, and this is assumed throughout Geometry and topology: "manifold"

Locally homeomorphic to Euclidean space means^[3] that every point has a neighborhood homeomorphic to an open Euclidean n-ball,

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 dimensions. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Since dimension is a local invariant, each connected component has a fixed dimension. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of

Scheme-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on ) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry. In Mathematics, an analytic manifold is a Topological manifold with analytic transition maps Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

Broad definition

The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. This omits the point-set axioms (allowing higher cardinalities and non-Hausdorff manifolds) and finite dimension (allowing various manifolds from functional analysis). In Mathematics, it is a usual axiom of a Manifold to be a Hausdorff space, and this is assumed throughout Geometry and topology: "manifold" For functional analysis as used in psychology see the Functional analysis (psychology article Usually one relaxes one or the other condition: manifolds without the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures For functional analysis as used in psychology see the Functional analysis (psychology article

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. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. 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 properly represent the entire Earth. 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. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. For a topological manifold, the simple space is some Euclidean space Rⁿ and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Polar coordinates, for example, form a chart for the plane R² minus the positive x-axis and the origin. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by Another example of a chart is the map χ_top mentioned in the section above, a chart for the circle.

Atlases

The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple ways using different combinations of charts.

The atlas containing all possible charts consistent with a given atlas is called the maximal atlas. Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though it is useful for definitions, it is a very 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 Rⁿ to the manifold and then back to another (or perhaps the same) open ball in Rⁿ. 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 the 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 Rⁿ (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane For symplectic manifolds, the transition functions must be symplectomorphisms. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically

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 pseudogroups. In Mathematics, a pseudogroup is an extension of the group concept but one that grew out of the geometric approach of Sophus Lie, rather than out of

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

The chart maps the part of the sphere with positive z coordinate to a disc.

Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R² 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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R³:

The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R². 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

χ(x,y,z) = (x,y),

maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is 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 of points which are mapped to each other by transition maps. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X 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. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

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 ¹⁄_s on the first copy (the point t = 0 is not identified with any point on the first copy). This gives a circle.

Intrinsic and extrinsic view

The first construction and this construction are very similar, but they represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group 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 or normal to some surface through that point. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation.

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.

n-Sphere as a patchwork

The n-sphere Sⁿ is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. An n-sphere Sⁿ can be constructed by gluing together two copies of Rⁿ. The transition map between them is defined as

This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability 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. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved. In the mathematical disciplines of Topology and Geometric group theory, an orbifold (for "orbit-manifold" is a generalization of a Manifold. In Topology, a CW complex is a type of Topological space introduced by J Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space

One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts on the manifold. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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 and real projective spaces (starting with a plane and a sphere, respectively). In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, real projective space, or RP n is the Projective space of lines in R n +1

Cartesian products

The Cartesian product of manifolds is also a manifold. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. Not every manifold can be written as a product of other manifolds.

The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology, and a Cartesian product of charts is a chart for the product manifold. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural 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 cylinders, for example, as S¹ × S¹ and S¹ × [0, 1], respectively. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis

A finite cylinder is a manifold with boundary.

Manifold with boundary

A manifold with boundary is a manifold with an edge. For example a sheet of paper with rounded corners is a 2-manifold with a 1-dimensional boundary. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The edge of an n-manifold is an (n-1)-manifold. A disk (circle plus interior) is a 2-manifold with boundary. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. Its boundary is a circle, a 1-manifold. A ball (sphere plus interior) is a 3-manifold with boundary. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric Its boundary is a sphere, a 2-manifold. (See also Boundary (topology)). For a different notion of boundary related to Manifolds see that article

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 {(x₁, x₂, …, x_n) | Σ x_i² < 1}. Every boundary point has a neighborhood homeomorphic to the "half" n-ball {(x₁, x₂, …, x_n) | Σ x_i² < 1 and x₁ ≥ 0}. The homeomorphism must send the boundary point to a point with x₁ = 0.

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 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 [0, 1] × [0, 1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics This article is about the mathematical object See Mobius Band (music group for the music group

Classes of manifolds

For more details on this topic, see Categories of manifolds. In Mathematics, specifically Geometry and topology, there are many different notions of Manifold, with more or less structure and corresponding notions of"map

Topological manifolds

For more details on this topic, see topological manifold. In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below

The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rⁿ. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology, a local homeomorphism is a map from one Topological space to another that respects locally the topological structure of the two spaces This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rⁿ. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function These homeomorphisms are the charts of the manifold.

It is to be noted that 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 and Riemannian manifolds discussed below. In particular, a 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 and second countable. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability "

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. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of 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. Other authors allow disjoint unions of topological manifolds with differing dimensions to be called manifolds.

Differentiable manifolds

For more details on this topic, see differentiable manifold. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus.

For most applications a special kind of topological manifold, 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 on a differentiable manifold. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Each point of an n-dimensional differentiable manifold has a tangent space. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since

Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is infinitely differentiable. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic (they can be expressed as power series, which are essentially polynomials of infinite degree). This article is about both real and complex analytic functions 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. In Mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult

Riemannian manifolds

For more details on this topic, see Riemannian manifolds. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M

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 〈⋅,⋅〉 in a manner which varies smoothly from point to point. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. Given two tangent vectors u and v, the inner product 〈u,v〉 gives a real number. The dot (or scalar) product is a typical example of an inner product. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R This allows one to define various notions such as length, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields. Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

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 n-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it.

Finsler manifolds

For more details on this topic, see Finsler manifold. In Mathematics, particularly Differential geometry, a Finsler manifold is a Differentiable manifold M with a Banach norm defined over

A Finsler manifold allows the definition of distance, but not of angle; it is an analytic manifold in which each tangent space is equipped with a norm, ||·||, in a manner which varies smoothly from point to point. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used to define an inner product. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set.

Any Riemannian manifold is a Finsler manifold.

Lie groups

For more details on this topic, see Lie group. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps. Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

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 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 modulus 1 with multiplication as the group operation. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of n by n matrices with non-zero determinant. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation If the matrix entries are real numbers, this will be an n²-dimensional disconnected manifold. In Mathematics, the real numbers may be described informally in several different ways 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. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. Further examples can be found in the table of Lie groups. This article gives a table of some common Lie groups and their associated Lie algebras The following are noted the topological properties of the group ( Dimension

Other types of manifolds

• A complex manifold is a manifold modeled on Cⁿ with holomorphic transition functions on chart overlaps. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional Note that an n-dimensional complex manifold has dimension 2n as a real differentiable manifold.
• A CR manifold is a manifold modeled on boundaries of domains in Cⁿ. In Mathematics, a CR manifold is a Differentiable manifold together with a geometric structure modeled on that of a real Hypersurface in a Complex
• Infinite dimensional manifolds: to allow for infinite dimensions, one may consider Banach manifolds which are locally homeomorphic to Banach spaces. In Mathematics, a Banach manifold is a Manifold modeled on Banach spaces. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis Similarly, Fréchet manifolds are locally homeomorphic to Fréchet spaces. This article deals with Fréchet spaces in functional analysis
• A symplectic manifold is a kind of manifold which is used to represent the phase spaces in classical mechanics. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects They are endowed with a 2-form that defines the Poisson bracket. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition A closely related type of manifold is a contact manifold. In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle

Classification and invariants

For more details on this topic, see Classification of manifolds. In Mathematics, specifically Geometry and topology, the classification of manifolds is a basic question about which much is known and many open questions remain

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 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. In Mathematics, 4-manifold is a 4-dimensional Topological manifold. In Mathematics, the uniformization theorem for Surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. This entry describes the solution of the Poincaré conjecture at a level intended for the general public In Mathematics, specifically in Topology, surgery theory is the name given to a collection of techniques used to produce one Manifold from another in a This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. In Mathematics, specifically Geometry and topology, the classification of manifolds is a basic question about which much is known and many open questions remain 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. In Mathematics, a complete set of invariants for a classification problem is a collection of maps f_i: X \to Y_i \ (where X

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 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. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance.

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, no program exists that can decide whether two manifolds are diffeomorphic. In Mathematics, specifically Geometry and topology, the classification of manifolds is a basic question about which much is known and many open questions remain

Smooth manifolds have a rich set of invariants, coming from point-set topology, classic algebraic topology, and geometric topology. In Mathematics, specifically Geometry and topology, the classification of manifolds is a basic question about which much is known and many open questions remain In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant, also detected by homology) and genus (a homological invariant). A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Mathematics, genus has a few different but closely related meanings Topology Orientable surface

Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection. In Mathematics, specifically Differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described In Differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a Moving frame around a curve In the mathematical field of Differential geometry, an affine connection is a geometrical object on a Smooth manifold which connects nearby Tangent This distinction between no local invariants and local invariants is a common way to distinguish between geometry and topology. In Mathematics, geometry and topology is an Umbrella term for Geometry and Topology, as the line between these two is often blurred most All invariants of a smooth closed manifold are thus global.

Algebraic topology is a source of a number of important global invariant properties. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Some key criteria include the simply connected property and orientability (see below). In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be Indeed several branches of mathematics, such as homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics, a characteristic class is a way of associating to each Principal bundle on a Topological space X a Cohomology class

Examples of 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 Rⁿ. Given an ordered basis for Rⁿ, 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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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 orientable manifolds. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back 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, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in 3-space, and (3) the real projective plane, which arises naturally in geometry. This article is about the mathematical object See Mobius Band (music group for the music group In Mathematics, the Klein bottle is a certain non- orientable Surface, i Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

Möbius strip

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. This article is about the mathematical object See Mobius Band (music group for the music group 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.

Klein bottle

The Klein bottle immersed in three-dimensional space.

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. In Mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. In Mathematics, the Klein bottle is a certain non- orientable Surface, i 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. Note that 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 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. Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" 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, or the "number of handles" present in a surface. In Mathematics, genus has a few different but closely related meanings Topology Orientable 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. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

Generalizations of manifolds

• Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of "singularities" in the topology. In the mathematical disciplines of Topology and Geometric group theory, an orbifold (for "orbit-manifold" is a generalization of a Manifold. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be Roughly speaking, it is a space which locally looks like the quotients of some simple space (e. g. Euclidean space) by the actions of various finite groups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a finite group is a group which has finitely many elements 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. In Mathematics, a singular point of an Algebraic variety V is a point P that is 'special' (so singular in the geometric sense that V 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 subset of Euclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. Both are related to manifolds, but are constructed algebraically using sheaves instead of atlases. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

Because of singular points, a variety is in general not a manifold, though linguistically the French variété, German Mannigfaltigkeit and English manifold are largely synonymous. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be This article deals with the general meaning of the term "synonym" In French an algebraic variety is called une variété algébrique (an algebraic variety), while a smooth manifold is called une variété différentielle (a differential variety).

• CW-complexes: A CW complex is a topological space formed by gluing disks of different dimensionality together. In Topology, a CW complex is a type of Topological space introduced by J In general the resulting space is singular, and hence not a manifold. However, they are of central interest in algebraic topology, especially in homotopy theory, as they are easy to compute with and singularities are not a concern. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

Notes

1. ^ The notion of a map can formalized as a cell decomposition.
2. ^ In the narrow sense of requiring point-set axioms and finite dimension.
3. ^ Formally, locally homeomorphic means that each point m in the manifold M has a neighborhood homeomorphic to a neighborhood in Euclidean space, not to the unit ball specifically. However, given such a homeomorphism, the pre-image of an ε-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 manifolds, as the complex unit ball is not holomorphic to complex space. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

See also

• 3-manifold
• 4-manifold
• 5-manifold
• List of manifolds
• Surface (2-manifold)

References

• Freedman, Michael H. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions In Mathematics, 4-manifold is a 4-dimensional Topological manifold. In Mathematics, a 5-manifold is a 5-dimensional Topological manifold, possibly with a piecewise linear or smooth structure. This is a list of particular Manifolds by Wikipedia page See also List of geometric topology topics. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. , and Quinn, Frank (1990) Topology of 4-Manifolds. Princeton University Press. ISBN 0-691-08577-3.
• Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2. Inspired by Milnor and commonly used in undergraduate courses.
• Hempel, John (1976) 3-Manifolds. Princeton University Press. ISBN 0-8218-3695-1.
• Hirsch, Morris, (1997) Differential Topology. Springer Verlag. ISBN 0-387-90148-5. 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.
• Kirby, Robion C. and Siebenmann, Laurence C. (1977) Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton University Press. ISBN 0-691-08190-5. A detailed study of the category of topological manifolds. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets
• Lee, John M. (2000) Introduction to Topological Manifolds. Springer-Verlag. ISBN 0-387-98759-2.
• ------ (2003) Introduction to Smooth Manifolds. Springer-Verlag. ISBN 0-387-95495-3.
• Massey, William S. (1977) Algebraic Topology: An Introduction. Springer-Verlag. ISBN 0-387-90271-6.
• Milnor, John (1997) Topology from the Differentiable Viewpoint. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential Princeton University Press. ISBN 0-691-04833-9.
• Spivak, Michael (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Michael David Spivak (*1940 in Queens, New York) is a Mathematician specializing in Differential geometry, an expositor of Mathematics HarperCollins Publishers. ISBN 0-8053-9021-9. The standard graduate text.
• Munkres, James R. (2000) Topology. Prentice Hall. ISBN 0-13-181629-2.
• Neuwirth, L. P. , ed. (1975) Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox. Princeton University Press. ISBN 978-0-691-08170-0.
• Riemann, Bernhard, Gesammelte mathematische Werke und wissenschaftlicher Nachlass, Sändig Reprint. ISBN 3-253-03059-8.
  • 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).