Manifold

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

Motivating examples

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 the unit circle, x² + y² = 1, where the y-coordinate is positive. 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 :
Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle:
Together, these parts cover the whole circle, and the four charts form an atlas for the circle.
The top and right charts, and respectively, overlap in their domain: their intersection lies in the quarter of the circle where both and -coordinates are positive. Both map this part into the interval, though differently. Thus a function can be constructed, which takes values from the co-domain of back to the circle using the inverse, followed by back to the interval. If a is any number in, then:
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
and
Here s is the slope of the line through the point at coordinates and the fixed pivot point ; similarly, t is the opposite of the slope of the line through the points at coordinates and. The inverse mapping from s to is given by
It can be confirmed that x² + y² = 1 for all values of s and t. These two charts provide a second atlas for the circle, with the transition map
.
Each chart omits a single point, either for s or for t, 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

The sphere is an example of a surface. The unit sphere of implicit equation
may be covered by an atlas of six charts: the plane divides the sphere into two half spheres, which may both be mapped on the disc by the projection on the 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, and therefore one needs atlases for covering the whole Earth surface.

Other curves

Manifolds do not need to be connected ; 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 .
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, whereas deleting a point from a line segment gives a space with at most two pieces; topological operations always preserve the number of pieces.

Definition

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 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 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 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 for some nonnegative integer.
This implies that either the point is an isolated point, or it has a neighborhood homeomorphic to the open ball
This implies also that every point has a neighborhood homeomorphic to
since is homeomorphic, and even diffeomorphic to any open ball in it.
The 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 . For example, the 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, each connected component has a fixed dimension.
Sheaf-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry.

Charts, atlases, and transition maps

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.

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 a subset of some Euclidean space 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 are all differentiable, allows us to do calculus on it. Polar coordinates, for example, form a chart for the plane 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.