*This page deals only with the mathematical term. See also: River Manifold.*

In mathematics, a

**manifold**is a topological space that looks locally like the "ordinary" Euclidean space

**R**

^{n}and is a Hausdorff space. An example is the surface of a sphere such as Earth, which is not a plane, but small patches of it are homeomorphic to (i.e., topologically equivalent to) patches of the Euclidean plane. To make precise the notion of "looks locally like" one uses local coordinate systems or

*charts*. A connected manifold has a definite

*dimension*, the number of coordinates needed in each local coordinate system. What follows below is a clean, contemporary mathematical treatment of manifolds; the foundational aspects of the subject were clarified during the 1930s, making precise intuitions dating back to around 1950, and developed through differential geometry and Lie group theory.

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. These manifolds are called *differentiable*. In order to measure lengths and angles, even more structure is needed: one defines *Riemannian manifolds* to recover these geometrical ideas.

Differentiable manifolds are used in mathematics to describe geometrical objects; they are also the most natural and general setting to study differentiability. In physics, differentiable manifolds serve as the phase space in classical mechanics and four dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity.

Table of contents |

2 Differentiable manifolds 3 Riemannian manifolds 4 Generalizations |

### Topological manifolds

A **topological n-manifold with boundary** is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of

`E`

^{ n}(Euclidean

`n`-space) or an open subset of the closed half of

`E`

^{ n}. The set of points which have an open neighbourhood homeomorphic to

`E`

^{ n}is called the

**interior**of the manifold; it is always non-empty. The complement of the interior, i.e. the set of points which have an open neighbourhood homeomorphic to a closed half of

`E`

^{ n}, is called the

**boundary**; it is an (

`n`-1)-manifold.

A manifold with empty boundary is said to be **closed** if it is compact, and **open** if it is not compact.

Manifolds inherit many of the local properties of Euclidean space.
In particular, they are locally path-connected, locally compact and locally metrizable.
(Readers should see the Topology Glossary for definitions of topological terms used in this article.)
Being locally compact Hausdorff spaces they are necessarily Tychonoff spaces. Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with *two* points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the *real line with two origins* is not Hausdorff, because the two origins cannot be separated.

Every connected manifold without boundary is homogeneous.

It can be shown that a manifold is metrizable if and only if it is paracompact.
Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an `n`-manifold.
Sometimes `n`-manifolds are defined to be second countable, which is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space.
Note that every compact manifold is second-countable, and every second-countable manifold is paracompact.

The classification of n-manifolds for n greater than four is known to be impossible; it is equivalent to the so-called word problem in group theory, which has been shown to be undecidable.

We know that every second-countable connected 1-manifold without boundary is homeomorphic either to **R** or the circle.
(The unconnected ones are just disjoint unions of these.)
For a classification of 2-manifolds, see Surface.

The 3-dimensional case is still open. Thurston's Geometrization Conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman may have proven this conjecture; his work is currently being evaluated, as of June 14, 2003.

### Differentiable manifolds

In order to discuss differentiability of functions, one needs more structure than a topological manifold provides. We start with a topological manifold `M` without boundary.
An open set of `M` together with a homeomorphism between the open set and an open set of `E`^{n} is called a coordinate **chart**.
A collection of charts which cover `M` is called an **atlas** of `M`.
The homeomorphisms of two overlapping charts provide a **transition** map from a subset of `E`^{n} to some other subset of `E`^{n}.
If all these maps are `k` times continuously differentiable, then the atlas is an ** C^{k} atlas**.

**Example:** The unit sphere in **R**^{3} can be covered by two charts: the complements of the north and south poles with coordinate maps - stereographic projections relative to the two poles.

Two `C`^{k} atlases are called *equivalent* if their union is a `C`^{k} atlas.
This is an equivalence relation, and a ** C^{k} manifold** is defined to be a manifold together with an equivalence class of

*atlases. If all the connecting maps are infinitely often differentiable, then one speaks of a*

`C`^{k}**smooth**or

**; if they are all analytic, then the manifold is an**

`C`^{∞}manifold**analytic**or

**.**

`C`^{ω}manifold
Intuitively, a smooth atlas provides local coordinate systems such that the change-of-coordinate functions are smooth.
These coordinate systems allow one to define differentiability and integrability of functions on `M`.

Associated with every point on a differentiable manifold is a tangent space and its dual, the cotangent space.
The former consists of the possible directional derivatives, and the latter of the differentials, which can be thought of as infinitesimal elements of the manifold.
These spaces always have the same dimension `n` as the manifold does.
The collection of all tangent spaces can in turn be made into a manifold, the tangent bundle, whose dimension is 2`n`.

If a `C`^{∞} manifold also carries a differentiable group structure, it is called a Lie group.
These are the proper objects for describing symmetries of analytical structures.

Once a `C`^{1} atlas on a paracompact manifold is given, we can refine it to a real analytic atlas (meaning that the new atlas, considered as a *C*^{1} atlas, is equivalent to the given one), and all such refinements give the same analytic manifold. Therefore, one often considers only these latter manifolds.

Not every topological manifold admits such a smooth atlas. The lowest dimension is 4 where there are non-smoothable topological manifolds. Also, it is possible for two non-equivalent differentiable manifolds to be homeomorphic. The famous example was given by John Milnor of wild 7-spheres, i.e. non-diffeomorphic topological 7-spheres.

### Riemannian manifolds

On differentiable manifolds, there are no notions of length, volume and angle. In order to introduce these, one needs a way to measure the lengths and angles between tangent vectors. A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion.

### Generalizations

The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. The diffeological spaces use a different notion of chart known as "plots". Differential spaces and Frölicher spaces are other attempts.

Manifolds "locally look like" Euclidean space **R**^{n} and are therefore inherently finite-dimensional objects. To allow for infinite dimensions, one may consider **Banach manifolds** which locally look like Banach spaces, or **Fréchet manifolds**, which locally look like Fréchet spaces.

See also Orbifold.