Manifold

For other uses, seeManifold (disambiguation).

TheKlein bottleimmersed in three-dimensional space

The surface of the Earth requires (at least) two charts to include every point. Here theglobeis decomposed into charts around theNorthandSouth Poles.

Inmathematics,amanifoldis atopological spacethat locally resemblesEuclidean spacenear each point. More precisely, an$n$-dimensional manifold, or$n$-manifoldfor short, is a topological space with the property that each point has aneighborhoodthat ishomeomorphicto anopen subsetof$n$-dimensional Euclidean space.

One-dimensional manifolds includelinesandcircles,but notself-crossing curves such as a figure 8.Two-dimensional manifolds are also calledsurfaces.Examples include theplane,thesphere,and thetorus,and also theKlein bottleandreal projective plane.

The concept of a manifold is central to many parts ofgeometryand modernmathematical physicsbecause it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets ofsystems of equationsand asgraphsof functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g.CT scans).

Manifolds can be equipped with additional structure. One important class of manifolds aredifferentiable manifolds;theirdifferentiable structureallowscalculusto be done. ARiemannian metricon a manifold allowsdistancesandanglesto be measured.Symplectic manifoldsserve as thephase spacesin theHamiltonian formalismofclassical mechanics,while four-dimensionalLorentzian manifoldsmodelspacetimeingeneral relativity.

The study of manifolds requires working knowledge of calculus andtopology.

Motivating examples

Circle

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

After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of theunit circle,x²+y²= 1, where they-coordinateis positive (indicated by the yellow arc inFigure 1). Any point of this arc can be uniquely described by itsx-coordinate. So,projectiononto the first coordinate is acontinuousandinvertiblemappingfrom the upper arc to theopen interval(−1, 1): $$\chi _{\mathrm {top} }(x,y)=x.\,$$

Such functions along with the open regions they map are calledcharts.Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle: $${\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}$$

Together, these parts cover the whole circle, and the four charts form anatlasfor the circle.

The top and right charts,$\chi _{\mathrm {top} }$and$\chi _{\mathrm {right} }$respectively, overlap in their domain: their intersection lies in the quarter of the circle where both$x$and$y$-coordinates are positive. Both map this part into the interval$(0,1)$,though differently. Thus a function$T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}$can be constructed, which takes values from the co-domain of$\chi _{\mathrm {top} }$back to the circle using the inverse, followed by$\chi _{\mathrm {right} }$back to the interval. Ifais any number in$(0,1)$,then: $${\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left[a\right]\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}$$

Such a function is called atransition 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 do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts $$\chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}$$ and $$\chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}$$

Heresis the slope of the line through the point at coordinates (x,y) and the fixed pivot point (−1, 0); similarly,tis the opposite of the slope of the line through the points at coordinates (x,y) and (+1, 0). The inverse mapping fromsto (x,y) is given by $${\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\[5pt]y&={\frac {2s}{1+s^{2}}}\end{aligned}}$$

It can be confirmed thatx²+y²= 1 for all values ofsandt.These two charts provide a second atlas for the circle, with the transition map $$t={\frac {1}{s}}$$ (that is, one has this relation betweensandtfor every point wheresandtare both nonzero).

Each chart omits a single point, either (−1, 0) forsor (+1, 0) fort,so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.

Sphere

Thesphereis an example of a surface. Theunit sphereofimplicit equation

x²+y²+z²– 1 = 0

may be covered by an atlas of sixcharts:the planez= 0divides the sphere into two half spheres (z> 0andz< 0), which may both be mapped on the discx²+y²< 1by the projection on thexyplane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes.

As with the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but the sphere cannot be covered by a single chart.

This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of theEarthcannot have a plane representation consisting of a singlemap(also called "chart", seenautical chart), and therefore one needsatlasesfor covering the whole Earth surface.

Other curves

Four manifolds fromalgebraic curves:■circles,■parabola,■hyperbola,■cubic.

Manifolds need not beconnected(all in "one piece" ); an example is a pair of separate circles.

Manifolds need not beclosed;thus a line segment without its end points is a manifold. They are nevercountable,unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are aparabola,ahyperbola,and thelocusof points on acubic curvey²=x³−x(a closed loop piece and an open, infinite piece).

However, excluded are examples like two touching circles that share a point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line segment, since deleting the center point from the "+" gives a space with fourcomponents(i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces;topological operationsalways preserve the number of pieces.

Mathematical definition

Further information:Categories of manifolds

Informally, a manifold is aspacethat is "modeled on" Euclidean space.

There are many different kinds of manifolds. Ingeometry and topology,all manifolds aretopological manifolds,possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions:homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance,differentiable manifoldshave homeomorphisms on overlapping neighborhoodsdiffeomorphicwith each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on the manifold as a whole.

Formally, a (topological) manifold is asecond countableHausdorff spacethat is locally homeomorphic to a Euclidean space.

Second countableandHausdorffarepoint-setconditions;second countableexcludes spaces which are in some sense 'too large' such as thelong line,whileHausdorffexcludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed innon-Hausdorff manifolds).

Locally homeomorphicto a Euclidean space means that every point has a neighborhoodhomeomorphicto anopen subsetof theEuclidean space$\mathbb {R} ^{n},$for some nonnegative integern.

This implies that either the point is anisolated point(if$n=0$), or it has a neighborhood homeomorphic to theopen ball $$\mathbf {B} ^{n}=\left\{(x_{1},x_{2},\dots,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1\right\}.$$This implies also that every point has a neighborhood homeomorphic to$\mathbb {R} ^{n}$ since$\mathbb {R} ^{n}$is homeomorphic, and evendiffeomorphicto any open ball in it (for$n>0$).

Thenthat appears in the preceding definition is called thelocal dimensionof the manifold. Generally manifolds are taken to have a constant local dimension, and the local dimension is then called thedimensionof the manifold. This is, in particular, the case when manifolds areconnected.However, some authors admit manifolds that are not connected, and where different points can have differentdimensions.If a manifold has a fixed dimension, this can be emphasized by calling it apure manifold.For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas thedisjoint unionof a sphere and a line in three-dimensional space isnota pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, islocally constant), eachconnected componenthas a fixed dimension.

Sheaf-theoretically,a manifold is alocally ringed space,whose structuresheafis 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 inalgebraic geometry.

Charts, atlases, and transition maps

Main article:Atlas (topology)

See also:Differentiable manifold

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

Charts

Main article:Coordinate chart

Acoordinate map,acoordinate chart,or simply achart,of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure.For a topological manifold, the simple space is a subset of some Euclidean space$\mathbb {R} ^{n}$and interest focuses on the topological structure. This structure is preserved byhomeomorphisms,invertible maps that are continuous in both directions.

In the case of a differentiable manifold, a set ofchartscalled anatlas,whosetransition functions(see below) are all differentiable, allows us to do calculus on it.Polar coordinates,for example, form a chart for the plane$\mathbb {R} ^{2}$minus the positivex-axis and the origin. Another example of a chart is the map χ_topmentioned above, a chart for the circle.

Atlases

Main article:Atlas (topology)

The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called anatlas.An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union is also an atlas.

The atlas containing all possible charts consistent with a given atlas is called themaximal atlas(i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations).

Transition maps

Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Russia may both contain Moscow. Given two overlapping charts, atransition functioncan be defined which goes from an open ball in$\mathbb {R} ^{n}$to the manifold and then back to another (or perhaps the same) open ball in$\mathbb {R} ^{n}$.The resultant map, like the mapTin the circle example above, is called achange of coordinates,acoordinate transformation,atransition function,or atransition map.

Additional structure

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure, the structure transfers to the manifold.

This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of$\mathbb {R} ^{n}$(that is, if they arediffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold.Complex manifoldsare introduced in an analogous way by requiring that the transition functions of an atlas areholomorphic functions.Forsymplectic manifolds,the transition functions must besymplectomorphisms.

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 calledcompatible.

These notions are made precise in general through the use ofpseudogroups.

Manifold with boundary

See also:Topological manifold § Manifolds with boundary

Amanifold with boundaryis a manifold with an edge. For example, a sheet of paper is a2-manifoldwith a 1-dimensional boundary. The boundary of an$n$-manifold with boundary is an$(n-1)$-manifold. Adisk(circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a1-manifold.Asquarewith interior is also a 2-manifold with boundary. Aball(sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (Do not confuse withBoundary (topology)).

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$\{(x_{1},x_{2},\dots,x_{n})\vert \Sigma x_{i}^{2}<1\}$.Every boundary point has a neighborhood homeomorphic to the "half"$n$-ball$\{(x_{1},x_{2},\dots,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}$.Any homeomorphism between half-balls must send points with$x_{1}=0$to points with$x_{1}=0$.This invariance allows to "define" boundary points; see next paragraph.

Boundary and interior

Let$M$be a manifold with boundary. Theinteriorof$M$,denoted$\operatorname {Int} M$,is the set of points in$M$which have neighborhoods homeomorphic to an open subset of$\mathbb {R} ^{n}$.Theboundaryof$M$,denoted$\partial M$,is thecomplementof$\operatorname {Int} M$in$M$.The boundary points can be characterized as those points which land on the boundary hyperplane$(x_{n}=0)$of$\mathbb {R} _{+}^{n}$under some coordinate chart.

If$M$is a manifold with boundary of dimension$n$,then$\operatorname {Int} M$is a manifold (without boundary) of dimension$n$and$\partial M$is a manifold (without boundary) of dimension$n-1$.

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 positivezcoordinate 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$\mathbb {R} ^{2}$is identified, and then an atlas covering this subset is constructed. The concept ofmanifoldgrew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:

Sphere with charts

A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of$\mathbb {R} ^{3}$: $$S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.$$

The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of$\mathbb {R} ^{2}$.Consider the northern hemisphere, which is the part with positivezcoordinate (coloured red in the picture on the right). The functionχdefined by $$\chi (x,y,z)=(x,y),\$$

maps the northern hemisphere to the openunit discby projecting it on the (x,y) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (x,z) plane and two charts projecting on the (y,z) plane, an atlas of six charts is obtained which covers the entire sphere.

This can be easily generalized to higher-dimensional spheres.

Patchwork

Further information:Surgery theory

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 anequivalence classof points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold.

This can be illustrated with the transition mapt=¹⁄_sfrom the second half of the circle example. Start with two copies of the line. Use the coordinatesfor the first copy, andtfor the second copy. Now, glue both copies together by identifying the pointton the second copy with the points=¹⁄_ton the first copy (the pointst= 0 ands= 0 are not identified with any point on the first and second copy, respectively). This gives a circle.

Intrinsic and extrinsic view

The first construction and this construction are very similar, but represent rather different points of view. In the first construction, the manifold is seen asembeddedin some Euclidean space. This is theextrinsic 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 istangentialornormalto some surface through that point.

The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called theintrinsic view.It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of anormal bundle,but instead there is an intrinsicstable normal bundle.

n-Sphere as a patchwork

Then-sphereSⁿis a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. Ann-sphereSⁿcan be constructed by gluing together two copies of$\mathbb {R} ^{n}$.The transition map between them isinversion in a sphere,defined as $$\mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.$$

This function is its own inverse and thus can be used in both directions. As the transition map is asmooth function,this atlas defines a smooth manifold. In the casen= 1, the example simplifies to the circle example given earlier.

Identifying points of a manifold

Main articles:OrbifoldandGroup action (mathematics)

It is possible to define different points of a manifold to be the same point. This can be visualized as gluing these points together in a single point, forming aquotient space.There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds,orbifoldsandCW complexesare considered to be relativelywell-behaved.An example of a quotient space of a manifold that is also a manifold is thereal projective space,identified as a quotient space of the corresponding sphere.

One method of identifying points (gluing them together) is through a right (or left) action of agroup,whichactson the manifold. Two points are identified if one is moved onto the other by some group element. IfMis the manifold andGis the group, the resulting quotient space is denoted byM/G(orG\M).

Manifolds which can be constructed by identifying points includetoriandreal projective spaces(starting with a plane and a sphere, respectively).

Gluing along boundaries

Main article:Quotient space (topology)

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 abijectionbetween 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. Aprojective planemay be obtained by gluing a sphere with a hole in it to aMöbius stripalong their respective circular boundaries.

Cartesian products

TheCartesian productof manifolds is also a manifold.

The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is theproduct 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 finitecylinders,for example, asS¹×S¹andS¹× [0,1], respectively.

A finite cylinder is a manifold with boundary.

History

Further information:History of manifolds and varieties

The study of manifolds combines many important areas of mathematics: it generalizes concepts such ascurvesand surfaces as well as ideas fromlinear algebraand topology.

Early development

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

Non-Euclidean geometryconsiders spaces whereEuclid'sparallel postulatefails.Saccherifirst studied such geometries in 1733, but sought only to disprove them.Gauss,BolyaiandLobachevskyindependently discovered them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise tohyperbolic geometryandelliptic geometry.In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positivecurvature,respectively.

Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. Histheorema egregiumgives a method for computing the curvature of asurfacewithout considering theambient spacein 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 anintrinsic 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 intrinsicpropertyof a manifold is itsEuler characteristic.Leonhard Eulershowed that for a convexpolytopein the three-dimensional Euclidean space withVvertices(or corners),Eedges, andFfaces,$$V-E+F=2.\$$The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating atopological mapwithVvertices,Eedges, andFfaces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope.Thus 2 is a topological invariant of the sphere, called itsEuler characteristic.On the other hand, atoruscan be sliced open by its 'parallel' and 'meridian' circles, creating a map withV= 1 vertex,E= 2 edges, andF= 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a usefultopological invariant,which can be extended to higher dimensions usingBetti numbers.In the mid nineteenth century, theGauss–Bonnet theoremlinked the Euler characteristic to theGaussian curvature.

Synthesis

Investigations ofNiels Henrik AbelandCarl Gustav Jacobion inversion ofelliptic integralsin the first half of 19th century led them to consider special types of complex manifolds, now known asJacobians.Bernhard Riemannfurther contributed to their theory, clarifying the geometric meaning of the process ofanalytic continuationof functions of complex variables.

Another important source of manifolds in 19th century mathematics wasanalytical mechanics,as developed bySiméon Poisson,Jacobi, andWilliam Rowan Hamilton.The possible states of a mechanical system are thought to be points of an abstract space,phase spaceinLagrangianandHamiltonianformalisms 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 theirgeneralized coordinates.For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but variousconservation lawsconstrain it to more complicated formations, e.g.Liouville tori.The theory of a rotating solid body, developed in the 18th century by Leonhard Euler andJoseph-Louis Lagrange,gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized byHenri Poincaré,one of the founders of topology.

Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The namemanifoldcomes from Riemann's originalGermanterm,Mannigfaltigkeit,whichWilliam Kingdon Cliffordtranslated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as aMannigfaltigkeit,because the variable can havemanyvalues. He distinguishes betweenstetige MannigfaltigkeitanddiskreteMannigfaltigkeit(continuous manifoldnessanddiscontinuous 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. Usinginduction,Riemann constructs ann-fach ausgedehnte Mannigfaltigkeit(n times extended manifoldnessorn-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of aMannigfaltigkeitevolved into what is today formalized as a manifold. Riemannian manifolds andRiemann surfacesare named after Riemann.

Poincaré's definition

In his very influential paper,Analysis Situs,Henri Poincaré gave a definition of a differentiable manifold (variété) which served as a precursor to the modern concept of a manifold.

In the first section of Analysis Situs, Poincaré defines a manifold as thelevel setof acontinuously differentiablefunction between Euclidean spaces that satisfies the nondegeneracy hypothesis of theimplicit function theorem.In the third section, he begins by remarking that thegraphof a continuously differentiable function is a manifold in the latter sense. He then proposes a new, more general, definition of manifold based on a 'chain of manifolds' (une chaîne des variétés).

Poincaré's notion of achain of manifoldsis a precursor to the modern notion of atlas. In particular, he considers two manifolds defined respectively as graphs of functions$\theta (y)$and$\theta '\left(y'\right)$.If these manifolds overlap (a une partie commune), then he requires that the coordinates$y$depend continuously differentiably on the coordinates$y'$and vice versa ('...les$y$sont fonctions analytiques des$y'$et inversement'). In this way he introduces a precursor to the notion of achartand of atransition map.

For example, the unit circle in the plane can be thought of as the graph of the function${\textstyle y={\sqrt {1-x^{2}}}}$or else the function${\textstyle y=-{\sqrt {1-x^{2}}}}$in a neighborhood of every point except the points (1, 0) and (−1, 0); and in a neighborhood of those points, it can be thought of as the graph of, respectively,${\textstyle x={\sqrt {1-y^{2}}}}$and${\textstyle x=-{\sqrt {1-y^{2}}}}$.The circle can be represented by a graph in the neighborhood of every point because the left hand side of its defining equation$x^{2}+y^{2}-1=0$has nonzero gradient at every point of the circle. By theimplicit function theorem,everysubmanifoldof Euclidean space is locally the graph of a function.

Hermann Weylgave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of atopological spacethat followed shortly. During the 1930sHassler Whitneyand others clarified thefoundationalaspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed throughdifferential geometryandLie grouptheory. Notably, theWhitney embedding theoremshowed that the intrinsic definition in terms of charts was equivalent to Poincaré's definition in terms of subsets of Euclidean space.

Topology of manifolds: highlights

Two-dimensional manifolds, also known as a 2Dsurfacesembedded in our common 3D space, were considered by Riemann under the guise ofRiemann surfaces,and rigorously classified in the beginning of the 20th century byPoul HeegaardandMax Dehn.Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as thePoincaré conjecture.After nearly a century,Grigori Perelmanproved the Poincaré conjecture (see theSolution of the Poincaré conjecture).William Thurston'sgeometrization 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 byMichael Freedmanand in a different setting, bySimon Donaldson,who was motivated by the then recent progress in theoretical physics (Yang–Mills theory), where they serve as a substitute for ordinary 'flat'spacetime.Andrey Markov Jr.showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higher-dimensional manifolds, includinganalogues of the Poincaré conjecture,had been done earlier byRené Thom,John Milnor,Stephen SmaleandSergei Novikov.A very pervasive and flexible technique underlying much work on thetopology of manifoldsisMorse theory.

Additional structure

Main article:Categories of manifolds

Topological manifolds

Main article:topological manifold

The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space$\mathbb {R} ^{n}$.By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological spacelocally homeomorphicto a Euclidean space. This means that every point has a neighbourhood for which there exists ahomeomorphism(abijectivecontinuous functionwhose inverse is also continuous) mapping that neighbourhood to$\mathbb {R} ^{n}$.These homeomorphisms are the charts of the manifold.

Atopologicalmanifold 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 anyparticularandconsistentchoice of such concepts.In order to discuss such properties for a manifold, one needs to specify further structure and considerdifferentiable manifoldsandRiemannian manifoldsdiscussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles.

Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space beHausdorffandsecond countable.

Thedimensionof the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (numbernin the definition). All points in aconnectedmanifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension.

Differentiable manifolds

Main article:Differentiable manifold

For most applications, a special kind of topological manifold, namely, adifferentiable 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 usecalculuson a differentiable manifold. Each point of ann-dimensional differentiable manifold has atangent space.This is ann-dimensional Euclidean space consisting of thetangent vectorsof the curves through the point.

Two important classes of differentiable manifolds aresmoothandanalytic 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 areanalytic(they can be expressed aspower series). The sphere can be given analytic structure, as can most familiar curves and surfaces.

Arectifiable setgeneralizes the idea of a piecewise smooth orrectifiable curveto higher dimensions; however, rectifiable sets are not in general manifolds.

Riemannian manifolds

Main article:Riemannian manifold

To measure distances and angles on manifolds, the manifold must be Riemannian. ARiemannian manifoldis a differentiable manifold in which eachtangent spaceis equipped with aninner product$\langle \cdot,\cdot \rangle$in a manner which varies smoothly from point to point. Given two tangent vectors$u$and$v$,the inner product$\langle u,v\rangle$gives a real number. Thedot(or scalar) product is a typical example of an inner product. This allows one to define various notions such as length,angles,areas(orvolumes),curvatureanddivergenceofvector fields.

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

Finsler manifolds

Main article:Finsler manifold

AFinsler manifoldallows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with anorm,$\|\cdot \|$,in a manner which varies smoothly from point to point. This norm can be extended to ametric,defining the length of a curve; but it cannot in general be used to define an inner product.

Any Riemannian manifold is a Finsler manifold.

Lie groups

Main article:Lie group

Lie groups,named afterSophus Lie,are differentiable manifolds that carry also the structure of agroupwhich is such that the group operations are defined by smooth maps.

A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of acompactLie group is the circle: the group operation is simply rotation. This group, known as$\operatorname {U} (1)$,can be also characterised as the group ofcomplex numbersofmodulus1 with multiplication as the group operation.

Other examples of Lie groups include special groups ofmatrices,which are all subgroups of thegeneral linear group,the group of$n\times n$matrices with non-zero determinant. If the matrix entries arereal numbers,this will be an$n^{2}$-dimensional disconnected manifold. Theorthogonal groups,thesymmetry groupsof the sphere andhyperspheres,are$n(n-1)/2$dimensional manifolds, where$n-1$is the dimension of the sphere. Further examples can be found in thetable of Lie groups.

Other types of manifolds

• Acomplex manifoldis a manifold whose charts take values in$\mathbb {C} ^{n}$and whose transition functions areholomorphicon the overlaps. These manifolds are the basic objects of study incomplex geometry.A one-complex-dimensional manifold is called aRiemann surface.An$n$-dimensional complex manifold has dimension$2n$as a real differentiable manifold.
• ACR manifoldis a manifold modeled on boundaries of domains in$\mathbb {C} ^{n}$.
• 'Infinite dimensional manifolds': to allow for infinite dimensions, one may considerBanach manifoldswhich are locally homeomorphic toBanach spaces.Similarly, Fréchet manifolds are locally homeomorphic toFréchet spaces.
• Asymplectic manifoldis a kind of manifold which is used to represent the phase spaces inclassical mechanics.They are endowed with a2-formthat defines thePoisson bracket.A closely related type of manifold is acontact manifold.
• Acombinatorial manifoldis a kind of manifold which is discretization of a manifold. It usually means apiecewise linear manifoldmade bysimplicial complexes.
• Adigital manifoldis a special kind of combinatorial manifold which is defined in digital space. Seedigital topology.

Classification and invariants

Further information: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 understoodin principle,except indimension 4:in low dimensions (2 and 3) it is geometric, via theuniformization theoremand thesolution of the Poincaré conjecture,and in high dimension (5 and above) it is algebraic, viasurgery theory.This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions.

Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms acomplete set of invariants.

This is much harder in higher dimensions: higher-dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object.

However, one can determine if two manifolds aredifferentif there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to asinvariants,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 areinvariantunder different descriptions.

One could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. It is known that for manifolds of dimension 4 and higher,no program existsthat can decide whether two manifolds are diffeomorphic.

Smooth manifolds havea rich set of invariants,coming frompoint-set topology,classic algebraic topology, andgeometric topology.The most familiar invariants, which are visible for surfaces, areorientability(a normal invariant, also detected byhomology) andgenus(a homological invariant).

Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably thecurvature of a Riemannian manifoldand thetorsionof a manifold equipped with anaffine connection.This distinction between local invariants and no local invariants is a common way to distinguish betweengeometry and topology.All invariants of a smooth closed manifold are thus global.

Algebraic topologyis a source of a number of important global invariant properties. Some key criteria include thesimply connectedproperty and orientability (see below). Indeed, several branches of mathematics, such ashomologyandhomotopytheory, and the theory ofcharacteristic classeswere founded in order to study invariant properties of manifolds.

Surfaces

Orientability

Main article:Orientable manifold

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$\mathbb {R} ^{n}$.Given anordered basisfor$\mathbb {R} ^{n}$,a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these areorientablemanifolds. 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) theMöbius strip,which is a manifold with boundary, (2) theKlein bottle,which must intersect itself in its 3-space representation, and (3) thereal projective plane,which arises naturally in geometry.

Möbius strip

Möbius strip

Main article: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 asingleside. Similarly to the Klein Bottle below, this two dimensional surface would need to intersect itself in two dimensions, but can easily be constructed in three or more dimensions.

See also:Quasitoric manifold

Klein bottle

Main article:Klein bottle

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

Real projective plane

Main article:Real projective space

Thereal projective planeis a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here asBoy's surface.

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

Genus and the Euler characteristic

For two dimensional manifolds a key invariant property is thegenus,or "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed, it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion ofEuler characteristic,and more generallyBetti numbersandhomologyandcohomology.

Maps of manifolds

AMorin surface,animmersionused insphere eversion

Main article:Maps of manifolds

Just as there are various types of manifolds, there are various types ofmaps of manifolds.In addition to continuous functions and smooth functions generally, there are maps with special properties. Ingeometric topologya basic type areembeddings,of whichknot theoryis a central example, and generalizations such asimmersions,submersions,covering spaces,andramified covering spaces. Basic results include theWhitney embedding theoremandWhitney immersion theorem.

In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions ofisometric embeddings,isometric immersions,andRiemannian submersions;a basic result is theNash embedding theorem.

Scalar-valued functions

3D color plot of thespherical harmonicsof degree$n=5$

A basic example of maps between manifolds are scalar-valued functions on a manifold, $$f\colon M\to \mathbb {R}$$ or $$f\colon M\to \mathbb {C},$$

sometimes calledregular functionsorfunctionals,by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold.

In geometric topology, most commonly studied areMorse functions,which yieldhandlebodydecompositions, while inmathematical analysis,one often studies solution topartial differential equations,an important example of which isharmonic analysis,where one studiesharmonic functions:the kernel of theLaplace operator.This leads to such functions as thespherical harmonics,and toheat kernelmethods of studying manifolds, such ashearing the shape of a drumand some proofs of theAtiyah–Singer index theorem.

Generalizations of manifolds

Infinite dimensional manifolds

The definition of a manifold can be generalized by dropping the requirement of finite dimensionality. Thus an infinite dimensional manifold is a topological space locally homeomorphic to atopological vector spaceover the reals. This omits the point-set axioms, allowing higher cardinalities andnon-Hausdorff manifolds;and it omits finite dimension, allowing structures such asHilbert manifoldsto be modeled onHilbert spaces,Banach manifoldsto be modeled onBanach spaces,andFréchet manifoldsto be modeled onFréchet spaces.Usually one relaxes one or the other condition: manifolds with the point-set axioms are studied ingeneral topology,while infinite-dimensional manifolds are studied infunctional analysis.

Orbifolds

Anorbifoldis 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 theactionsof variousfinite 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-singularalgebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subsets of Euclidean space, analgebraic varietyis glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields.Schemesare likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically usingsheavesinstead of atlases.

Because ofsingular points,a variety is in general not a manifold, though linguistically the Frenchvariété,GermanMannigfaltigkeitand Englishmanifoldare largelysynonymous.In French an algebraic variety is calledunevariété algébrique(analgebraic variety), while a smooth manifold is calledunevariété différentielle(adifferential variety).

Stratified space

A "stratified space" is a space that can be divided into pieces ( "strata" ), with each stratum a manifold, with the strata fitting together in prescribed ways (formally, afiltrationby closed subsets).There are various technical definitions, notably a Whitney stratified space (seeWhitney conditions) for smooth manifolds and atopologically stratified spacefor topological manifolds. Basic examples includemanifold with boundary(top dimensional manifold and codimension 1 boundary) and manifolds with corners (top dimensional manifold, codimension 1 boundary, codimension 2 corners). Whitney stratified spaces are a broad class of spaces, including algebraic varieties, analytic varieties,semialgebraic sets,andsubanalytic sets.

CW-complexes

ACW complexis a topological space formed by gluing disks of different dimensionality together. In general the resulting space is singular, hence not a manifold. However, they are of central interest in algebraic topology, especially inhomotopy theory.

Homology manifolds

Ahomology manifoldis a space that behaves like a manifold from the point of view of homology theory. These are not all manifolds, but (in high dimension) can be analyzed by surgery theory similarly to manifolds, and failure to be a manifold is a local obstruction, as in surgery theory.

Differential spaces

Let$M$be a nonempty set. Suppose that some family of real functions on$M$was chosen. Denote it by$C\subseteq \mathbb {R} ^{M}$.It is an algebra with respect to the pointwise addition and multiplication. Let$M$be equipped with the topology induced by$C$.Suppose also that the following conditions hold. First: for every$H\in C^{\infty }\left(\mathbb {R} ^{n}\right)$,where$n\in \mathbb {N}$,and arbitrary$f_{1},\dots,f_{n}\in C$,the composition$H\circ \left(f_{1},\dots,f_{n}\right)\in C$.Second: every function, which in every point of$M$locally coincides with some function from$C$,also belongs to$C$.A pair$(M,C)$for which the above conditions hold, is called a Sikorski differential space.

See also

• Geodesic– Straight path on a curved surface or a Riemannian manifold
• Directional statistics– subdiscipline of statisticsPages displaying wikidata descriptions as a fallback:statistics on manifolds
• List of manifolds
• Timeline of manifolds– Mathematics timeline
• Mathematics of general relativity– Mathematical structures and techniques used in the theory of general relativity

By dimension

• 3-manifold– Mathematical space
• 4-manifold– Mathematical space
• 5-manifold– Manifold of dimension five
• Manifolds of mappings– locally convex vector spaces satisfying a very mild completeness conditionPages displaying wikidata descriptions as a fallback

References

• Freedman, Michael H.,and Quinn, Frank (1990)Topology of 4-Manifolds.Princeton University Press.ISBN0-691-08577-3.
• Guillemin, Victorand Pollack, Alan (1974)Differential Topology.Prentice-Hall.ISBN0-13-212605-2.Advanced undergraduate / first-year graduate text inspired by Milnor.
• Hempel, John (1976)3-Manifolds.Princeton University Press.ISBN0-8218-3695-1.
• Hirsch, Morris,(1997)Differential Topology.Springer Verlag.ISBN0-387-90148-5.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.ISBN0-691-08190-5.A detailed study of thecategoryof topological manifolds.
• Lee, John M. (2000)Introduction to Topological Manifolds.Springer-Verlag.ISBN0-387-98759-2.Detailed and comprehensive first-year graduate text.
• Lee, John M. (2003)Introduction to Smooth Manifolds.Springer-Verlag.ISBN0-387-95495-3.Detailed and comprehensive first-year graduate text; sequel toIntroduction to Topological Manifolds.
• Massey, William S.(1977)Algebraic Topology: An Introduction.Springer-Verlag.ISBN0-387-90271-6.
• Milnor, John(1997)Topology from the Differentiable Viewpoint.Princeton University Press.ISBN0-691-04833-9.Classic brief introduction to differential topology.
• Munkres, James R.(1991)Analysis on Manifolds.Addison-Wesley (reprinted by Westview Press)ISBN0-201-51035-9.Undergraduate text treating manifolds in$\mathbb {R} ^{n}$.
• Munkres, James R.(2000)Topology.Prentice Hall.ISBN0-13-181629-2.
• Neuwirth, L. P., ed. (1975)Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox.Princeton University Press.ISBN978-0-691-08170-0.
• Riemann, Bernhard,Gesammelte mathematische Werke und wissenschaftlicher Nachlass,Sändig Reprint.ISBN3-253-03059-8.
  • 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.W.A. Benjamin Inc. (reprinted by Addison-Wesley and Westview Press).ISBN0-8053-9021-9.Famously terseadvanced undergraduate / first-year graduate text.
• Spivak, Michael(1999)A Comprehensive Introduction to Differential Geometry(3rd edition) Publish or Perish Inc. Encyclopedic five-volume series presenting a systematic treatment of the theory of manifolds, Riemannian geometry, classical differential geometry, and numerous other topics at the first- and second-year graduate levels.
• Tu, Loring W. (2011).An Introduction to Manifolds(2nd ed.). New York: Springer.ISBN978-1-4419-7399-3..Concise first-year graduate text.

External links

• "Manifold",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Dimensions-math.org(A film explaining and visualizing manifolds up to fourth dimension.)
• Themanifold atlasproject of theMax Planck Institute for Mathematics in Bonn