Manifold

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

A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, different kinds of manifold can be constructed, such as topological manifolds and differentiable manifolds. A mathematical definition of these structures follows the motivational example below. Applications of manifolds to physics include differentiable manifolds which serve as the phase space in classical mechanics and four-dimensional pseudo-Riemannian manifolds which are used to model spacetime in general relativity.

Introduction

A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or three dimensional spaces: a line, a plane, or the three-dimensional space in which we live. Or, it may be an abstract space of some higher dimension or even of infinite dimension. Some authors allow manifolds to have separate pieces of different dimensions, but all authors require all pieces of a connected manifold to have the same dimension. A manifold with all pieces of dimension n is called an n-manifold. By contrast, gluing a one-dimensional string to three dimensional ball makes an object called a CW complex, not a manifold.

People once thought the Earth was flat, an understandable illusion since we are very small compared with the Earth. So too, an ideal mathematical sphere is like a plane in any sufficiently small region. This means that a sphere is a 2-manifold. However, a sphere and a plane have quite different global structure: if one walks over the surface of a sphere in a fixed direction, one eventually returns to the starting point, whereas on an infinite plane, one would continue forever.

An example of a one-dimensional manifold is a loop of string. A two-dimensional manifold is also called a surface. One example is a sphere. In higher dimensions, manifolds become harder to visualize, but are very important in physics where, for example, the set of rotations in three-dimensional space is a three-dimensional manifold or 3-manifold. From this example, we see that a manifold can be an abstract space, which may be hard to picture.

There are many different classes of manifolds. The simplest are topological manifolds, which look locally like some Euclidean space. Other classes of manifolds have additional structure. One of the most important kind of manifold is the differentiable manifold, which has a structure that permits the application of calculus.

We navigate the spherical Earth using flat maps or charts, collected in an atlas. Similarly, we can describe a differentiable manifold 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 holistically 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.

The idea of a Riemannian manifold, a differentiable manifold on which distances can be defined, 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 part of the circle, for which the y-coordinate is positive (this is the yellow part in Figure 1). Any point in this part can be described by the x-coordinate. So, there is a continuous 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) parts 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 charts overlap. Their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χ_top and χ_right map this part bijectively to the interval (0, 1). Thus we can form a function T from (0, 1) to itself by first inverting the yellow chart to reach the circle and then following 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

\chi _{\mathrm {plus} }(x,y)=t={y \over {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 topological manifold examples 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.

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.

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 negative x-axis and the origin. Another example of a chart is the map χ_top mentioned in the section above, a chart for the circle.

Atlases

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

The atlas containing all possible charts consistent with a given atlas is called the maximal atlas. Unlike an ordinary atlas, the maximal atlas of a given atlas 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, we can define a transition function which goes from an open ball in Rⁿ to the manifold and then back to another (or perhaps the same) open ball in Rⁿ. The resultant map, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map.

Additional structure

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

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

In 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, thereby leading to a slightly different viewpoint.

Charts

Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R² is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:

Sphere with charts

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.

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.

This can be illustrated with the transition map t = ¹⁄_s from the second half of the circle example. Start with two copies of the line. Use the coordinate s for the first copy, and t for the second copy. Now, glue both copies together by identifying the point t on the second copy with the point ¹⁄_s on the first copy (the point t = 0 is not identified with any point on the first copy). This gives a circle.

Intrinsic and extrinsic view

The first construction and this construction are very similar, but they represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. 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.

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 makes it much harder to imagine what a tangent vector might be.

n-Sphere as a patchwork

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.

If we take n = 1, we retrieve the example with the circle.

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 as zeros of a function

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.

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

Cartesian products

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. Cartesian products may be used to construct tori and finite cylinders, for example, as S¹ × S¹ and S¹ × [0, 1], respectively.

Gluing along boundaries

Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together.

Formally, the gluing is defined by a bijection 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 R × [0, 1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A Klein bottle may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries.

Topological manifolds

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.

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 dimension of the Euclidean space 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 the same Euclidean space. In that case every topological manifold has a topological invariant, its dimension. Other authors allow disjoint unions of topological manifolds with differing dimensions.

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.

Orientability

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 we can think of 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, we can choose charts 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.

We consider three examples: (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.

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 we will perform a little "surgery". 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. We now have 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.

Klein bottle

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

Real projective plane

Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called antipodes. Although we have no way to do so physically, we 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 projects to the same "point" on this "plane".

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 related to manifolds, but are constructed using sheaves instead of atlases.

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 required a generalization of manifolds to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translates as "manifoldness". In his Göttingen inaugural lecture, Riemann states the possible values a property can attain form a Mannigfaltigkeit. He distinguishes between stetige Mannigfaltigkeit and discrete [sic] 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 or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldness. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds, Riemann surfaces, and Riemann sums are named after Riemann; each is useful in mathematical analysis.

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. All these use the notion of several characteristic axes or dimensions, but these dimensions do not lie along the physical dimensions of width, height, and breadth.

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 3-sphere? As of 2005, a consensus amongst experts is developing that recent work by Grigori Perelman may have disposed of this question, after nearly a century.

Hermann Weyl gave an intrinsic definition for differentiable 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 Pollack, Anton, Differential Topology, Prentice-Hall (1974), ISBN 0132126052. This text was inspired by Milnor, and is commonly used in undergraduate courses.
Hirsch, Morris, Differential Topology, Springer (1997), ISBN 0387901485. Hirsch provides the most complete account with historical insights and excellent, but difficult, problems. This is the standard reference for those wishing to have a deep understanding of the subject.
Kirby, Robion C. and 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.
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.