This page deals only with the mathematical term. See also: River Manifold.
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 and one defines Riemannian manifolds.
Differentiable manifolds are used in mathematics to describe geometrical objects; they are also the most natural and general setting to study differentiability (but see diffeology for more general notions). 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. What follows is a clean mathematical treatment of 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. 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.
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.
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 En 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 En to some other subset of En. If all these maps are k times continuously differentiable, then the atlas is an Ck atlas.
Example: The unit sphere in R3 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 Ck atlases are called equivalent if their union is a Ck atlas. This is an equivalence relation, and a Ck manifold is defined to be a manifold together with an equivalence class of Ck atlases. If all the connecting maps are infinitely often differentiable, then one speaks of a smooth or C∞ manifold; if they are all analytic, then the manifold is an analytic or Cω 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 2n.
Once a C1 atlas on a paracompact manifold is given, we can refine it to a real analytic atlas (meaning that the new atlas, considered as a C1 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.
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.
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 Rn 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.