A manifoldMathworldPlanetmath is a space that is locally like n, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties, such as non-contractible loops (, that distinguish it from the topologically trivial n.

Standard Definition.

An n-dimensional topological manifold M is a second countable, Hausdorff topological space11For connectedPlanetmathPlanetmath manifolds, the assumptionPlanetmathPlanetmath that M is second-countable is logically equivalent to M being paracompact, or equivalently to M being metrizable. The topological hypotheses in the definition of a manifold are needed to exclude certain counter-intuitive pathologies. Standard illustrations of these pathologies are given by the long line (lack of paracompactness) and the forked line (points cannot be separated). These pathologies are fully described in Spivak. See this page ( that is locally homeomorphic to open subsets of n.

A differential manifold is a topological manifold with some additional structureMathworldPlanetmath information. A chart, also known as a system of local coordinates, is a mapping α:Un, such that the domain UM is an open set, and such that U is homeomorphic to the image α(U). Let α:Uαn, and β:Uβn be two charts with overlapping domains ( The continuousMathworldPlanetmathPlanetmath injection


is called a transition functionMathworldPlanetmath, and also called a a change of coordinates. An atlas 𝒜 is a collectionMathworldPlanetmath of charts α:Uαn whose domains cover M, i.e.


Note that each transition function is really just n real-valued functions of n real variables, and so we can ask whether these are continuously differentiable. The atlas 𝒜 defines a differential structure on M, if every transition function is continuously differentiable.

More generally, for k=1,2,,,ω, the atlas 𝒜 is said to define a 𝒞k differential structure, and M is said to be of class 𝒞k, if all the transition functions are k-times continuously differentiable, or real analytic in the case of 𝒞ω. Two differential structures of class 𝒞k on M are said to be isomorphicPlanetmathPlanetmath if the union of the corresponding atlases is also a 𝒞k atlas, i.e. if all the new transition functions arising from the merger of the two atlases remain of class 𝒞k. More generally, two 𝒞k manifolds M and N are said to be diffeomorphic, i.e. have equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath differential structure, if there exists a homeomorphism ϕ:MN such that the atlas of M is equivalent to the atlas obtained as ϕ-pullbacks of charts on N.

The atlas allows us to define differentiable mappings to and from a manifold. Let


be a continuous function. For each α𝒜 we define


called the representation of f relative to chart α, as the suitably restricted compositionMathworldPlanetmath


We judge f to be differentiableMathworldPlanetmath if all the representations fα are differentiable. A path


is judged to be differentiable, if for all differentiable functions f, the suitably restricted composition fγ is a differentiable function from to . Finally, given manifolds M,N, we judge a continuous mapping ϕ:MN between them to be differentiable if for all differentiable functions f on N, the suitably restricted composition fϕ is a differentiable function on M.

Title manifold
Canonical name Manifold
Date of creation 2013-03-22 12:20:22
Last modified on 2013-03-22 12:20:22
Owner matte (1858)
Last modified by matte (1858)
Numerical id 35
Author matte (1858)
Entry type Definition
Classification msc 53-00
Classification msc 57R50
Classification msc 58A05
Classification msc 58A07
Synonym differentiable manifold
Synonym differential manifold
Synonym smooth manifold
Related topic NotesOnTheClassicalDefinitionOfAManifold
Related topic LocallyEuclidean
Related topic 3Manifolds
Related topic Surface
Related topic TopologicalManifold
Related topic ProofOfLagrangeMultiplierMethodOnManifolds
Related topic SubmanifoldMathworldPlanetmath
Defines coordinate chart
Defines chart
Defines local coordinates
Defines atlas
Defines change of coordinates
Defines differential structure
Defines transition function
Defines smooth structure
Defines diffeomorphism
Defines diffeomorphic
Defines topological manifold
Defines real-analytic manifold