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Encyclopedia > Manifold
On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). 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.

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. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

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. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... For other uses, see Calculus (disambiguation). ... In Riemannian geometry, a Riemannian manifold is a real 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, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... Phase space of a dynamical system with focal stability. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...

### Circle

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

$chi_{mathrm{top}}(x,y) = x . ,!$

Such functions along with the open regions they map are called charts. 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. In topology, a branch of mathematics, an atlas describes how a complicated space called a manifold is glued together from simpler pieces. ...

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 χtop and χright 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. Let a be any number in (0,1), then: A function Æ’ and its inverse Æ’â€“1. ...

begin{align} T(a) &= chi_{mathrm{right}}left(chi_{mathrm{top}}^{-1}(a)right) &= chi_{mathrm{right}}left(a, sqrt{1-a^2}right) &= sqrt{1-a^2} . end{align}

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 Image File history File links Download high resolution version (1000x1000, 75 KB) File links The following pages link to this file: User talk:KSmrq Manifold/rewrite ... Image File history File links Download high resolution version (1000x1000, 75 KB) File links The following pages link to this file: User talk:KSmrq Manifold/rewrite ...

$chi_{mathrm{minus}}(x,y) = s = frac{y}{1+x}$

and

$chi_{mathrm{plus}}(x,y) = t = frac{y}{1-x}.$

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

begin{align} x &= frac{1-s^2}{1+s^2} y &= frac{2s}{1+s^2} . end{align}

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

$t = frac{1}{s} . ,!$

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. They need not be closed; thus a line segment without its end points is a manifold. And they need not be finite; thus a parabola is a manifold. 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 = x3x (a closed loop piece and an open, infinite piece). Image File history File links Download high resolution version (1000x1000, 42 KB) PNG file created as SVG, rendered by Batik, and uploaded by author. ... Image File history File links Download high resolution version (1000x1000, 42 KB) PNG file created as SVG, rendered by Batik, and uploaded by author. ... Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ... In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. ...

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. The transition map T, and all the others, are differentiable on (0, 1); therefore, with this atlas the circle is a differentiable manifold. It is also smooth and analytic because the transition functions have these properties as well. For other uses, see Calculus (disambiguation). ... For other uses, see Derivative (disambiguation). ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ...

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 is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...

## 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 algebra and topology. 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, a surface is a two-dimensional manifold. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...

### Prehistory

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

Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied them in 1733. Lobachevsky, Bolyai, and Riemann developed 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. Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... For other uses, see Euclid (disambiguation). ... a and b are parallel, the transversal t produces congruent angles. ... Giovanni Gerolamo Saccheri (September 5, 1667 â€“ October 25, 1733) was an Italian Jesuit priest and mathematician. ... Events February 12 - British colonist James Oglethorpe founds Savannah, Georgia. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (ÐÐ¸ÐºÐ¾Ð»Ð°ÌÐ¹ Ð˜Ð²Ð°ÌÐ½Ð¾Ð²Ð¸Ñ‡ Ð›Ð¾Ð±Ð°Ñ‡ÐµÌÐ²ÑÐºÐ¸Ð¹) (December 1, 1792â€“February 24, 1856 (N.S.); November 20, 1792â€“February 12, 1856 (O.S.)) was a Russian mathematician. ... JÃ¡nos Bolyai (December 15, 1802â€“January 27, 1860) was a Hungarian mathematician. ... Bernhard Riemann. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Lines through a given point P and asymptotic to line l. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... In Riemannian geometry, a Riemannian manifold is a real 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, 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. His theorema egregium gives a method for computing the curvature of a surface 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 property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Johann Carl Friedrich Gauss (pronounced ,  ; in German usually GauÃŸ, Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... The Theorema Egregium (Remarkable Theorem) is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... An open surface with X-, Y-, and Z-contours shown. ... The ambient space, in mathematics, is the space surrounding a mathematical object. ...

Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges, and F faces, A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ... Euler redirects here. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...

V-E+F= 2.

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.[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. 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. For other uses, see Sphere (disambiguation). ... A torus This article is about the surface and mathematical concept of a torus. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... 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 the space into two pieces. ... The Gaussâ€“Bonnet theorem or Gaussâ€“Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ...

### 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. 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. Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ... Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... Bernhard Riemann. ... 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. 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. 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 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. Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton. ... Simeon Poisson. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ... Phase space of a dynamical system with focal stability. ... 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 invented in 1833 by William Rowan Hamilton. ... 2-dimensional renderings (ie. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Euler redirects here. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ... Jules Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in 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 term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Bernhard Riemann. William Kingdon Clifford William Kingdon Clifford, FRS (May 4, 1845 - March 3, 1879) was an English mathematician who also wrote a fair bit on philosophy. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Bernhard Riemann. ...

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. 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. Hermann Klaus Hugo Weyl (November 9, 1885 â€“ December 9, 1955) was a German mathematician. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Hassler Whitney (23 March 1907 â€“ 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

### 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. Henri 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 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). Bill Thurston's 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' space-time. 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. One of the most pervasive and flexible techniques underlying much work on the topology of manifolds is Morse theory. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Poul Heegaard (November 2, 1871 — February 7, 1948) was a mathematician active in the field of topology. ... Max Dehn (November 13, 1878 â€“ June 27, 1952) was a German mathematician. ... Jules Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... In mathematics, the PoincarÃ© conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... Grigori Yakovlevich Perelman (Russian: ), born 13 June 1966 in Leningrad, USSR (now St. ... William Paul Thurston (born October 30, 1946) is an American mathematician. ... Thurstons geometrization conjecture states that compact 3-manifolds can be decomposed into pieces with geometric structures. ... Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, USA) is a mathematician at Microsoft Research. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is an English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... RenÃ© Thom (September 2, 1923 - October 25, 2002) was a French mathematician and founder of the catastrophe theory. ... John Willard Milnor (b. ... Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. ... Sergei Petrovich Novikov (also Serguei) (Russian: Ð¡ÐµÑ€Ð³ÐµÐ¹ ÐŸÐµÑ‚Ñ€Ð¾Ð²Ð¸Ñ‡ ÐÐ¾Ð²Ð¸ÐºÐ¾Ð²) (born 20 March 1938) is a Russian mathematician, noted for work in both algebraic topology and soliton theory. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A Morse function is also an expression for an anharmonic oscillator In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ...

## Mathematical definition

For more details on this topic, see Categories of manifolds.

Informally, a manifold is a space that is "modeled on" Euclidean space. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

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 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. Geometry and Topology (ISSN 1364-0380 online, 1465-3060 printed) is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

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. Specifically, a space is said to be second countable if its topology has a countable base. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... Topological spaces are structures that allow one to formalize 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 that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In topology, the long line is a topological space analogous to the real line, but much longer. ...

Locally homeomorphic to Euclidean space means[3] that every point has a neighborhood homeomorphic to an open Euclidean n-ball, This word should not be confused with homomorphism. ...

$mathbf{B}^n = { (x_1, x_2, dots, x_n)inmathbb{R}^n mid x_1^2 + x_2^2 + cdots + x_n^2 < 1 }.$

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. Since dimension is a local invariant, each connected component has a fixed dimension. 2-dimensional renderings (ie. ... Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...

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.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... 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 the language and the problematics of geometry. ...

The broadest common definition of 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 finite dimension (allowing various manifolds from functional analysis). 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 a topological vector space is one of the basic structures investigated in functional analysis. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

## 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 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. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

### 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 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. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... This word should not be confused with homomorphism. ...

In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. Polar coordinates, for example, form a chart for the plane R2 minus the positive x-axis and the origin. Another example of a chart is the map χtop mentioned in the section above, a chart for the circle. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

### 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.

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. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ...

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 abstract algebra (such as quasigroup, for example). ...

## 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: Image File history File links A sphere with the chart mapping the upper hemisphere to a disk. ... Image File history File links A sphere with the chart mapping the upper hemisphere to a disk. ...

#### 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 R3: For other uses, see Sphere (disambiguation). ...

$S = { (x,y,z) in mathbf{R}^3 | x^2 + y^2 + z^2 = 1 }.$

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. 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. A disc of unit radius on a plane is called a unit disc. ...

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. 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. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

This can be illustrated with the transition map t = 1s 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 1s 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. 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. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... A normal vector is a vector which is perpendicular to a surface or manifold. ...

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. An n-sphere Sn can be constructed by gluing together two copies of Rn. The transition map between them is defined as 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ...

$mathbf{R}^n setminus {0} to mathbf{R}^n setminus {0}: x mapsto x/|x|^2.$

This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold. In the case n = 1, the example simplifies to the circle example given earlier. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...

### 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, orbifolds and CW complexes are considered to be relatively well-behaved. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In topology and 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.H.C. Whitehead to meet the needs of homotopy theory. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ...

One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts 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). This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a symmetry group describes all symmetries of objects. ...

Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a plane and a sphere, respectively). A torus This article is about the surface and mathematical concept of a torus. ... In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ...

### Cartesian products

The Cartesian product of manifolds is also a manifold. Not every manifold can be written as a product of other manifolds. In mathematics, the Cartesian product is a direct product of sets. ...

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. 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. In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ...

A finite cylinder is a manifold with boundary.

Image File history File links Download high resolution version (711x641, 30 KB) Right circular cylinder, created in Matlab by Jitse Niesen. ... Image File history File links Download high resolution version (711x641, 30 KB) Right circular cylinder, created in Matlab by Jitse Niesen. ...

### 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. The edge of an n-manifold is an (n-1)-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (See also Boundary (topology)). In geometry, a disk is the region in a plane contained inside of a circle. ... 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 spaces in general. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...

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, …, xn) | Σ xi2 < 1}. Every boundary point has a neighborhood homeomorphic to the "half" n-ball {(x1, x2, …, xn) | Σ xi2 < 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. Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... A MÃ¶bius strip made with a piece of paper and tape. ...

## Classes of manifolds

For more details on this topic, see Categories of manifolds.

### Topological manifolds

For more details on this topic, see topological manifold.

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. 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. These homeomorphisms are the charts of the manifold. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Topological spaces are structures that allow one to formalize 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. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... A bijective function. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

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 is a topological space in which points can be separated by neighbourhoods. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ...

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. Other authors allow disjoint unions of topological manifolds with differing dimensions to be called manifolds. Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...

### Differentiable manifolds

For more details on this topic, see differentiable manifold.

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. 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. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... For other uses, see Calculus (disambiguation). ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

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 (they can be expressed as power series, which are essentially polynomials of infinite degree). The sphere can be given analytic structure, as can most familiar curves and surfaces. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

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. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

### Riemannian manifolds

For more details on this topic, see Riemannian manifolds.

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.

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. 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 Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... 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.

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. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... This picture illustrates how the hours on a clock form a group under modular addition. ...

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. 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. If the matrix entries are real numbers, this will be an n2-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. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, the general linear group of degree n is the set of nÃ—n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... The symmetry group of an object (e. ... For other uses, see Sphere (disambiguation). ... 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ... This article gives a table of some common Lie groups and their associated Lie algebras. ...

### Other types of manifolds

• A complex manifold is a manifold modeled on Cn with holomorphic transition functions on chart overlaps. These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. 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.
• 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 is a kind of manifold which is used to represent the phase spaces in classical mechanics. They are endowed with a 2-form that defines the Poisson bracket. A closely related type of manifold is a contact manifold.

In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In mathematics, a Banach manifold is a manifold modeled on Banach spaces. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... This article deals with FrÃ©chet spaces in functional analysis. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. ...

## Classification and invariants

For more details on this topic, see Classification of manifolds.

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. 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. 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 Gauss curvature. ... 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 controlled way. ...

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 (where X is the collection of objects being classified, up to some equivalence relation, and the are some sets), such that âˆ¼ if and only if for all i. ...

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. ...

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.

Smooth manifolds have 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 homology) and genus (a homological invariant). In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... The torus is an orientable surface. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...

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. This distinction between no local invariants and local invariants is a common way to distinguish between geometry and topology. All invariants of a smooth closed manifold are thus global. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ... In differential geometry, the torsion tensor is one of the tensorial invariants of a connection on the tangent bundle. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... Geometry and Topology (ISSN 1364-0380 online, 1465-3060 printed) is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. ...

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 and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... The two bold paths shown above are homotopic relative to their endpoints. ... In mathematics, the idea of characteristic class is one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry. ...

## 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. 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. 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. In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... The torus is an orientable surface. ...

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. A MÃ¶bius strip made with a piece of paper and tape. ... The Klein bottle immersed in three-dimensional space. ... The fundamental polygon of the projective plane. ... For other uses, see Geometry (disambiguation). ...

Möbius strip

Mobius strip created with Mathematica. ... Mobius strip created with Mathematica. ...

#### 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. A MÃ¶bius strip made with a piece of paper and tape. ...

#### 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. 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. 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. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a MÃ¶bius strip. ... The Klein bottle immersed in three-dimensional 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. 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". The fundamental polygon of the projective 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. 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, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ...

## Generalizations of manifolds

• Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of "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 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 subset of Euclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. 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 sheaves instead of atlases.
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 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 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.

In topology and 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 well-behaved in some particular way, such as differentiability. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a finite group is a group which has finitely many elements. ... 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 is not locally flat there. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... 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 well-behaved in some particular way, such as differentiability. ... Synonyms (in ancient Greek syn συν = plus and onoma όνομα = name) are different words with similar or identical meanings. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

## 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 such that every neighborhood looks like the complex n-space in a coherent way. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

This is a list of particular manifolds, by Wikipedia page. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, a 3-manifold is a 3-dimensional manifold. ... In mathematics, 4-manifold is a 4-dimensional topological manifold. ... There are very few or no other articles that link to this one. ...

## References

• Freedman, Michael H., 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.
• 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. Princeton University Press. ISBN 0-691-04833-9.
• 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).
• Spivak, Michael (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers. ISBN 0-8053-9021-9. The standard graduate text.

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... John Willard Milnor (b. ... Bernhard Riemann. ... Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ...

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