A manifold, in mathematics, can be thought of as a "curved" surface or space which locally looks like Euclidean space and therefore admits the introduction of local charts or coordinate systems. An example is the surface of the Earth, which is not flat but small patches of it are reasonably well approximated by patches of the Euclidean plane. Every manifold has a dimension, the number of coordinates needed in local coordinate systems. The dimension of the surface of the Earth is two. 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. In physics, differentiable manifolds serve as the phase space in classical mechanics; 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 is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open set of E n (Euclidean n-space) or an open set of a 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, and its complement is called the boundary. 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.
The classification of all manifolds is an open problem. We know that every second-countable connected 1-manifold without boundary is homeomorphic either to R or the circle. For a classification of 2-manifolds, see surface.
In order to discuss differentiability of functions, one needs more structure than a topological manifolds provides. We start with a topological manifold M without boundary. An open set of M together with a homeomorphism from the open set to an open set of En is called a chart. A collection of charts which cover M is called an atlas of M. The homeomorphisms from two overlapping charts provide a 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. 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 Ck atlas provides local coordinate systems such that the change-of-coordinate functions are k times continuously differentiable. These coordinate systems allow to define differentiability and integrability of functions on M.
Associated with every point on an Ck 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 as the manifold does. The collection of all tangent spaces can in turn be made into a manifold, the tangent bundle.