# manifold

## Summary.

A manifold^{} is a space that is
locally like ${\mathbb{R}}^{n}$, however lacking a preferred system of
coordinates. Furthermore, a manifold can have global topological
properties, such as non-contractible loops (http://planetmath.org/Curve), that distinguish it from
the topologically trivial ${\mathbb{R}}^{n}$.

## Standard Definition.

An $n$-dimensional topological manifold $M$ is a second countable, Hausdorff
topological space^{1}^{1}For connected^{} manifolds, the assumption^{} 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 (http://planetmath.org/BibliographyForDifferentialGeometry).
that is locally homeomorphic to open subsets of
${\mathbb{R}}^{n}$.

A differential manifold is a topological manifold with some additional
structure^{} information. A *chart*, also known as a *system
of local coordinates*, is a mapping $\alpha :U\to {\mathbb{R}}^{n}$, such that the domain $U\subset M$ is an open set, and such that $U$ is homeomorphic to the image $\alpha (U)$. Let
$\alpha :{U}_{\alpha}\to {\mathbb{R}}^{n},$ and
$\beta :{U}_{\beta}\to {\mathbb{R}}^{n}$ be two charts with overlapping
domains (http://planetmath.org/Function). The continuous^{} injection

$$\beta \circ {\alpha}^{-1}:\alpha ({U}_{\alpha}\cap {U}_{\beta})\to {\mathbb{R}}^{n}$$ |

is called a *transition function ^{}*,
and also called a

*a change of coordinates*. An atlas $\mathcal{A}$ is a collection

^{}of charts $\alpha :{U}_{\alpha}\to {\mathbb{R}}^{n}$ whose domains cover $M$, i.e.

$$M=\bigcup _{\alpha}{U}_{\alpha}.$$ |

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 $\mathcal{A}$ defines a differential structure on $M$, if every transition function is continuously differentiable.

More generally, for $k=1,2,\mathrm{\dots},\mathrm{\infty},\omega $, the atlas $\mathcal{A}$ is
said to define a ${\mathcal{C}}^{k}$ differential structure, and $M$ is said to be
of class ${\mathcal{C}}^{k}$, if all the transition functions are $k$-times
continuously differentiable, or real analytic in the case of
${\mathcal{C}}^{\omega}$. Two differential structures of class ${\mathcal{C}}^{k}$ on $M$ are
said to be isomorphic^{} if the union of the corresponding atlases is
also a ${\mathcal{C}}^{k}$ atlas, i.e. if all the new transition functions arising
from the merger of the two atlases remain of class ${\mathcal{C}}^{k}$. More
generally, two ${\mathcal{C}}^{k}$ manifolds $M$ and $N$ are said to be
diffeomorphic, i.e. have equivalent^{} differential structure, if there
exists a homeomorphism $\varphi :M\to N$ such that the atlas of $M$ is
equivalent to the atlas obtained as $\varphi $-pullbacks of charts on $N$.

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

$$f:U\to \mathbb{R},U\subset M$$ |

be a continuous function. For each $\alpha \in \mathcal{A}$ we define

$${f}_{\alpha}:V\to \mathbb{R},V\subset {\mathbb{R}}^{n},$$ |

called the
representation of $f$ relative to chart $\alpha $, as the suitably
restricted composition^{}

$${f}_{\alpha}=f\circ {\alpha}^{-1}.$$ |

We judge $f$ to be differentiable^{} if all
the representations ${f}_{\alpha}$ are differentiable. A path

$$\gamma :I\to M,I\subset \mathbb{R}$$ |

is judged to be differentiable, if for all differentiable functions $f$, the suitably restricted composition $f\circ \gamma $ is a differentiable function from $\mathbb{R}$ to $\mathbb{R}$. Finally, given manifolds $M,N$, we judge a continuous mapping $\varphi :M\to N$ between them to be differentiable if for all differentiable functions $f$ on $N$, the suitably restricted composition $f\circ \varphi $ 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 | Submanifold^{} |

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 |