Manifold

This is a proposed rewrite of manifold. Of course, all the deleted parts have to be found a home in some other article, such as Manifold/rewrite/topological manifold or Manifold/rewrite/differentiable manifold

Target audience (roughly):

For lead section and examples: high school.
For Topological manifolds and Smooth manifolds: first-year or second-year undergraduates.
For the rest: graduates.

You are welcome to edit it.

For other meanings of this term, see manifold (disambiguation).

In mathematics, a manifold generalizes the idea of a surface. Technically, it can be constructed using multiple overlapping pieces to form a whole and is, in this sense, like a patchwork. On a small scale manifolds are always simple; on a large scale, they have rich flexibility.

Much of the terminology is inspired by cartography, which uses flat drawings to depict features on the Earth, as in navigational charts and city maps. Thus, for example, we speak of an atlas of local charts.

In the remainder of this article we will give precise definitions and explore a few of the many and varied examples of manifolds.

Introduction

A manifold can be characterised as looking like Euclidean space, or some other relatively simple space, in a close-up view. For instance, people once considered the Earth flat, an understandable illusion since we are very small compared to the Earth. So, too, an ideal mathematical sphere is like a plane in any sufficiently tiny region, which makes it a manifold. However, a sphere and a plane have quite different global structure: if you walk over the surface of a sphere in a fixed direction, you eventually return to your starting point, whereas on a plane, you will continue forever.

A surface is two-dimensional. However, manifolds can have any dimension. Other examples include a closed loop of string (one-dimensional) and the set of rotations in three-dimensional space (three-dimensional). The example of rotation space also shows that a manifold can be an abstract space.

Local simplicity is a strong demand. For example, we cannot dangle a string from a sphere and call the whole a manifold; no region containing the point where the string is attached to the sphere is simple — either a line or a plane — however tiny.

We navigate on the Earth using flat maps, collected in an atlas. Similarly, we can describe a manifold using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is generally not possible to describe a manifold with just one chart, due to the differences in global structure between the manifold and the simple space on which it is modeled. When using multiple charts which together cover a manifold, we need to pay attention to the regions where they overlap, because it is the overlaps which encode the global structure.

There are many different kinds of manifolds. The simplest are topological manifolds, which look locally like some Euclidean space. Other variations come with additional structure needed in their use. For example, a differentiable manifold supports not just topology, but differential and integral calculus. The idea of a Riemannian manifold led to the mathematics of general relativity, describing a space-time continuum with curvature.

Motivational example: the circle

The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1 with its centre at the origin. If x and y are the coordinates of a point on the circle, then we have x² + y² = 1.

Locally, the circle resembles a line, which is one-dimensional. In other words, we need only one coordinate to describe the circle locally. Consider, for instance, the top half of the circle, for which the y-coordinate is positive (this is the yellow part in the figure to the right). Any point in this part can be described by the x-coordinate. So, there is a bijection χ_top, which maps the yellow part of the circle to the open interval (−1, 1) by simply projecting onto the first coordinate:

\chi _{\mathrm {top} }(x,y)=x.\,

Such a function is called a chart. Similarly, there are charts for the bottom (red), left (blue), and right (green) halves of the circle. Together, these parts cover the whole circle and we say that the four charts form an atlas for the circle.

Note that the top and right halves overlap. Their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χ_top and χ_right map this part bijectively to the interval (0, 1). Thus we can form a function T from (0, 1) to itself by first following the yellow chart to the circle and then the green chart back to the interval:

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}}}.

Such a function is called a transition map.

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

\chi _{\mathrm {minus} }(x,y)=s={y \over {1+x}}

and χ_plus(x,y) = t = y/(1−x). Here s is the slope of the line through the variable 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

x={{1-s^{2}} \over {1+s^{2}}},\qquad y={{2s} \over {1+s^{2}}};

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

t={1 \over s}.

Notice that 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. With tools from topology we can show that no single chart can ever cover the full circle; already in this simple example we require the flexibility of manifolds with their multiple charts.

Manifolds need not be connected (all in "one piece"); thus a pair of separate circles is also a topological manifold. They need not be closed; thus a line segment without its ends is a manifold. And they need not be finite; thus a parabola is a topological manifold. Putting these freedoms together, two other example topological manifolds are a hyperbola and the locus of points on the cubic curve y²−x³+x=0.

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 to one-dimensional Euclidean space. (We may take a different view in algebraic geometry, where we consider complex points on the quartic curve ((x−1)²+y²−1)((x+1)²+y²−1)=0, whose real points alone form a pair of circles touching at the origin.)

Viewed through the eyes of calculus, the circle transition function T is simply a function between open intervals, so we know what it means for T to be differentiable. In fact, T is differentiable on (0, 1) and the same goes for the other transition maps. Therefore, this atlas turns the circle into a differentiable manifold.

More examples

The following examples occur in various sections of this article:

n-Sphere: See n-sphere (patchwork), n-sphere (zeros of a function).
Cylinder: See cylinder (Cartesian product), cylider (gluing along boundaries).
Torus: See torus (identifying points of a manifold), torus (Cartesian product).
Klein bottle: See Klein bottle (gluing allong boundaries).
Möbius band: See Möbius band (orientation).
Line with two origins: See Line with two origins (Hausdorff).

Sphere

The surface of the sphere can be treated in almost the same way as the circle. We view the sphere as a subset of R³:

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 for instance the northern hemisphere, which is the part with positive z coordinate (it is coloured red in the picture on the right). The function χ, defined by

\chi (x,y,z)=(x,y),

maps the northern hemisphere to the open unit disc 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, we obtain an atlas of six charts covering the entire sphere.

This can be easily generalized to higher-dimensional spheres.

Charts, atlases and transition maps

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 we are interested in the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions.

A chart is especially useful for calculations, as it allows to calculate in the simple space and transport the result back to the manifold.

Polar coordinates, for example, form a chart for R² without the negative x-axis and the origin. The map χ_top mentioned in the section above is a chart for the circle.

Atlases

It is generally not possible to describe a manifold with just one chart, but several charts are needed to cover it. A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as a manifold can be covered using different charts.

The atlas containing all possible charts is called the maximal atlas. Unlike an ordinary atlas, the maximal atlas is unique. Though it is useful for definitions, it is a very abstract object and not used for calculations.

Transition maps

Charts in an atlas normally 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. A map between those parts correlating points which represent the same point in the manifold, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map.

Additional structure

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

This is the standard way differentiable manifolds are defined. If the transition maps of an atlas for a topological manifold preserve the natural differential structure of Rⁿ (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold.

In general the structure on the manifold depends on the atlas, but sometimes different atlases give rise to the same structure. Such atlases are called compatible.

Construction

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

Patchwork

A manifold can be constructed by gluing together overlapping pieces in a consistent manner. 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 on each other by transition maps and a chart maps points of a single patch to their equivalence classes. There are strong demands on the consistency of the transition maps. At least they are required to be homeomorphisms, giving rise to a topological manifold; if they are also diffeomorphisms, the resulting manifold is differentiable.

n-Sphere

The n-sphere Sⁿ can constructed by gluing together two copies of Rⁿ. The transition map between them is defined as

\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.

See also n-sphere (zeros of a function).

Zeros of a function

Many manifolds can be defined as the set of zeros of a specific function. This construction naturally embeds the manifold into a Euclidean space and thus leads to an extrinsic view. It is very graphic, but unfortunately not suitable for every manifold.

If the Jacobian matrix of a differentiable function has maximal rank at every point where the function is zero, then according to the implicit function theorem, there is a whole neighborhood of zeros around each such point that is diffeomorphic to a Euclidean space. Hence the set of zeros is a manifold.

n-Sphere

The n-sphere Sⁿ is often defined as

\mathbf {S} ^{n}:=\{x\in \mathbf {R} ^{n+1}:\|x\|=1\}

which is equivalent to the zeros of the function

x\mapsto \|x\|-1.

The Jacobian matrix of this function is

{\begin{bmatrix}x_{1}&\ldots &x_{n+1}\end{bmatrix}},

which has rank one (the maximum for a 1×n matrix) for all points but the origin. This proves that the n-sphere is a differentiable manifold.

See also n-sphere (patchwork).

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. Most often the result is not a manifold, but in some cases it is.

The identification in those cases is done using a group, which acts on the manifold. Two points are identified if one is moved on the other by some group element. If M is the manifold and G is the group, the resulting space is called the quotient space and denoted by M/G.

Torus

The group Z² acts on on the plane R² by translation. An element (a, b) of Z² moves a point (x, y) of R² to the point (x+a, y+b). The resulting quotient space R² / Z² is the torus.

See also torus (Cartesian product).

Projective plane

The group Z/2Z consisting of only elements 0 and 1 acts on the sphere by reflection, with 0 acting as the identity and 1 acting as the reflection at the origin, mapping a point to its antipode. The resulting quotient space S²/(Z/2Z) is the projective plane.

Cartesian product

The Cartesian product of manifolds is also a manifold. Not every manifold can be written as a product.

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.

Cylinder

The Cartesian product S¹ × R of a circle with the real line is the cylinder.

If a closed interval, say [0, 1], is used instead of the real line, the resulting manifold S¹ × [0, 1] is a finite cylinder and has a boundary consisting of two disjoint circles.

See also cylinder (gluing along boundaries).

Torus

The Cartesian product S¹ × S¹ of two circles is the torus. Since neither circle has a boundary, neither does the torus.

See also torus (identifying points).

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 also be glued together.

Formally, the gluing is defined by a homeomorphism between two components of the boundaries. Two points are identified when they are mapped onto each other by this homeomorphism. If this homeomorphism is also a diffeomorphism and the original manifold was differentiable, the resulting manifold is also a differentiable manifold. If it preserves any other structure, this structure also extends to the resulting manifold.

Cylinder

When the two edges of a strip are glued together the result is a cylinder.

The strip can be defined as the product manifold R × [0, 1]. Its boundary has two components, each one a copy of R, and

(x,0)\mapsto (x,1)

defines a diffeomorphism between them.

See also cylinder (Cartesian product).

Klein bottle

A sphere with a hole has one boundary component, a circle. This is also true of the Möbius strip with boundary. If the sphere and Möbius strip are glued together along these diffeomorphic boundaries, the result is the Klein bottle.

Intrinsic and extrinsic view

Every real differentiable manifold can be 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 an Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point. When we view a manifold simply as a topological space without any embedding, then it is much harder to imagine what a tangent vector might be. This is the intrinsic view.

If you imagine yourself, or an ant, within a certain manifold, say the surface of Earth, you have the intrinsic view. When you step outside of the manifold, say by getting into a rocket and flying into space, and then look back at the ant, you have the extrinsic view.

The circle can be defined intrinsically by gluing together two copies of the line. We do this by identifying non-zero points in the first copy by their multiplicative inverse in the second copy. This circle is not embedded in anything.

Topological manifolds

The simplest kind of manifold to define is the topological manifold, which looks locally like "ordinary" Euclidean space Rⁿ. Formally, a topological manifold is a topological space locally homeomorphic to 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 an open subset of Rⁿ. These homeomorphisms are the charts of the manifold.

Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space is Hausdorff and second countable. This means that the line with two origins, discussed above, is not a topological manifold since it is not Hausdorff.

The dimension of the manifold at a certain point is the number n in the definition. All points in a connected manifold have the same dimension. In that case, we call the manifold an n-manifold.

Differentiable manifolds

It is easy to define topological manifolds, but it is very hard to work with them. For most applications a special kind of topological manifold, a differentiable manifold, works better. If the local charts on a manifold are compatible in a certain sense, one can talk about directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to apply "calculus" on a differentiable manifold.

Orientation

Möbius band

The Möbius band M is constructed by gluing together two copies of R². Define

(\mathbf {R} ^{2})_{\mathrm {L} }:=\{x\in \mathbf {R} ^{2}:x_{1}<0\}\quad (\mathbf {R} ^{2})_{\mathrm {R} }:=\{x\in \mathbf {R} ^{2}:x_{1}>0\}

(\mathbf {R} ^{2})_{\mathrm {L} }\to (\mathbf {R} ^{2})_{\mathrm {L} }:x=(x_{1},x_{2})\mapsto (1/x_{1},x_{2})

(\mathbf {R} ^{2})_{\mathrm {R} }\to (\mathbf {R} ^{2})_{\mathrm {R} }:x=(x_{1},x_{2})\mapsto (1/x_{1},-x_{2})

Hausdorff

Line with two origins

This is a slightly more exotic example, in the sense that it is not a Hausdorff space. Take two copies of R, write them as $\mathbf {R} \times \{0\}$ and $\mathbf {R} \times \{1\}$ , and define an equivalence relation by

(x,0)\sim (x,1)

if

x\neq 0

.

The quotient space L obtained from this equivalence relation is a space like the real line, except two points "occupy" the origin. In particular, they cannot be separated by disjoint open sets, so L is non-Hausdorff. It is a manifold, however, in the most general sense, because each copy of R maps injectively into L, and the inverses of these maps provide charts.

Often, a manifold is defined to be a Hausdorff space, so examples such as this cannot occur.

Other types and generalizations of manifolds

In order to do geometry on manifolds it is usually necessary to adorn these spaces with additional structure, such as that described above for differentiable manifolds. There are numerous other possibilities, depending on the kind of geometry one is interested in:

Complex 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.

Banach and Fréchet 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.

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: 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 generalizations of manifolds.

History

The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intrinsic differential geometry with his theorema egregium.

Bernhard Riemann was the first to do extensive work that really required a generalization of manifolds to higher dimensions. The name manifold comes from Bernhard Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translates as "manifoldness". In his Göttingen inaugural lecture, Riemann states that the possible values a property can attain form a Mannigfaltigkeit. He distinguishes between stetige and discrete [sic] Mannigfaltigkeit (continuous and discontinuous manifoldness), depending on whether the value changes continuously or not. As examples for stetige Mannigfaltikeiten he mentions colors and the locations of objects in space, but also the possible shapes of a spatial figure. He constructs an n fach ausgedehnte Mannigfaltigkeit (n times extended or n-dimensional manifoldness) as a continuous stack of (n−1) fach ausgedehnte Mannigfaltigkeiten. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Riemann.

Abelian varieties were already implicitly known at Riemann's time, as complex manifolds. Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, are also naturally manifold theories.

Henri Poincaré studied three-dimensional manifolds and raised a question, nowadays known as the Poincaré conjecture: Are all closed, simply connected three-dimensional manifolds homeomorphic to the sphere? This question has not been fully resolved yet, but Grigori Perelman seems to be making good progress.

Hermann Weyl gave an intrinsic definition for differential manifolds in 1912. The foundational aspects of the subject were clarified during the 1930s by Hassler Whitney and others, making precise intuitions dating back to the latter half of the 19th century, and developed through differential geometry and Lie group theory.

References

Guillemin, Victor and Anton Pollack, Differential Topology, Prentice-Hall (1974) ISBN 0132126052. This text was inspired by Milnor, and is commonly used for undergraduate courses.
Hirsch, Morris, Differential Topology, Springer (1997) ISBN 0387901485. Hirsch gives the most complete account with historical insights and excellent, but difficult problems. This is the best reference for those wishing to have a deep understanding of the subject.
Kirby, Robion C.; Siebenmann, Laurence C. Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton, New Jersey: Princeton University Press (1977). ISBN 0-691-08190-5. A detailed study of the category of topological manifolds.
Lee, John M. Introduction to Topological Manifolds, Springer-Verlag, New York (2000). ISBN 0-387-98759-2. Introduction to Smooth Manifolds, Springer-Verlag, New York (2003). ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.
Milnor, John, Topology from the Differentiable Viewpoint, Princeton University Press, (revised, 1997) ISBN 0691048339. This short text may be the best math book ever written.
Spivak, Michael, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers (1965). ISBN 0805390219. This is the standard text used in most graduate courses.
Riemann, Bernhard, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. The 1851 doctoral thesis in which "manifold" (Mannigfaltigkeit) first appears.
Riemann, Bernhard, On the Hypotheses which lie at the Bases of Geometry. The famous Göttingen inaugural lecture (Habilitationsschrift) of 1854.