Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an \(n\)-dimensional manifold, or \(n\)-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of \(n\)-dimensional Euclidean space.

One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure-eight. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of 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 are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

The study of manifolds requires working knowledge of calculus and topology.

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 the unit circle, x + y = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous and invertible mapping from the upper arc to the open interval (−1, 1): \[\chi_{\mathrm{top}}(x,y) = x . \,\]

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

Together, these parts cover the whole circle, and the four charts form an atlas for 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 \(T:(0,1)\rightarrow(0,1)=\chi_\mathrm{right} \circ \chi^{-1}_\mathrm{top}\) 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. If a is 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 a transition map.

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 \[\chi_\mathrm{plus}(x, y) = t = \frac{y}{1 - x}\]

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

Condensed: the full section is in Wikipedia.

The sphere is an example of a surface. The unit sphere of implicit equation

x + y + z, 1 = 0

may be covered by an atlas of six charts: the plane z = 0 divides the sphere into two half spheres (z > 0 and z < 0), which may both be mapped on the disc x + y < 1 by the projection on the xy plane 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 the Earth cannot have a plane representation consisting of a single map (also called "chart", see nautical chart), and therefore one needs atlases for covering the whole Earth surface.

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

Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic curve y = x − 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 four components (i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces; topological operations always preserve the number of pieces.

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

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 transition maps: 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 manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on the manifold as a whole.

Formally, a (topological) manifold is a second countable Hausdorff space that is locally homeomorphic to a Euclidean space.

Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense 'too large' such as the long line, while Hausdorff excludes spaces such as the line with two origins.

Locally homeomorphic to a Euclidean space means that every point has a neighborhood homeomorphic to an open subset of the Euclidean space \(\R^n,\) for some nonnegative integer n.

This implies that either the point is an isolated point (if \(n=0\)), or it has a neighborhood homeomorphic to the open ball \[\mathbf{B}^n = \left\{ (x_1, x_2, \dots, x_n)\in\R^n: x_1^2 + x_2^2 + \cdots + x_n^2 < 1 \right\}.\] This implies also that every point has a neighborhood homeomorphic to \(\R^n\) since \(\R^n\) is homeomorphic, and even diffeomorphic to any open ball in it (for \(n>0\)).

The n that appears in the preceding definition is called the local dimension of the manifold. Generally manifolds are taken to have a constant local dimension, and the local dimension is then called the dimension of the manifold. This is, in particular, the case when manifolds are connected. However, some authors admit manifolds that are not connected, and where different points can have different dimensions. If a manifold has a fixed dimension, this can be emphasized by calling it a pure manifold. For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is not a 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, is locally constant), each connected component has a fixed dimension.

Condensed: the full section is in Wikipedia.

The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can 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.

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 a subset of some Euclidean space \(\R^n\) and interest focuses on 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, whose transition functions (see below) are all differentiable, allows us to do calculus on it. Polar coordinates, for example, form a chart for the plane \(\R^2\) minus the positive x-axis and the origin. Another example of a chart is the map χ_top mentioned above, a chart for the circle.

The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an atlas. 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 the maximal 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).

A manifold can be defined as a topological space equipped with an atlas (and in some definitions, a maximal atlas).

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, a transition function can be defined which goes from an open ball in \(\R^n\) to the manifold and then back to another (or perhaps the same) open ball in \(\R^n\). The resultant map, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map.

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all 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 \(\R^n\) (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms.

The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible.

These notions are made precise in general through the use of pseudogroups.

A manifold with boundary is a manifold with an edge. For example, a disk (circle plus interior) is a 2-manifold with boundary the circle, a 1-manifold. The boundary of an \(n\)-manifold with boundary is an \((n-1)\)-manifold. In three dimensions, a ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold.

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 \(\textstyle\{(x_1,x_2,\dots,x_n)\mid\sum x_i^2<1\}\). Every boundary point has a neighborhood homeomorphic to the "half" \(n\)-ball \(\textstyle\{(x_1,x_2,\dots,x_n)\mid\sum x_i^2<1\text{ and }x_1\geq0\}\). Any homeomorphism between half-balls must send points with \(x_1=0\) to points with \(x_1=0\). This invariance allows us to define boundary points on the manifold with boundary (see next paragraph).

If in addition we are considering differentiable manifolds, then a square with interior is not a manifold with boundary. The four vertices are neither locally diffeomorphic to Euclidean space nor to Euclidean half-space. This is an example of a differentiable manifold with corners. Similarly, products of differentiable manifolds with boundaries are not generally differentiable manifolds with boundaries, but instead are differentiable manifolds with corners.

Let \(M\) be a manifold with boundary. The interior of \(M\), denoted \(\operatorname{Int} M\), is the set of points in \(M\) which have neighborhoods homeomorphic to an open subset of \(\R^n\). The boundary of \(M\), denoted \(\partial M\), is the complement of \(\operatorname{Int} M\) in \(M\). The boundary points can be characterized as those points which land on the boundary hyperplane \((x_n=0)\) of \(\R^n_+\) 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 \(n-1\).

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

Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of \(\R^2\) 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: