manifold
Summary.
Standard Definition.
An -dimensional topological manifold is a second countable, Hausdorff
topological space11For connected manifolds, the assumption
that is
second-countable is logically equivalent to being paracompact, or
equivalently to 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
.
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 , such that the domain is an open set, and such that is homeomorphic to the image . Let
and
be two charts with overlapping
domains (http://planetmath.org/Function). The continuous
injection
is called a transition function,
and also called a a change of coordinates. An atlas
is a collection
of charts
whose domains cover , i.e.
Note that each transition function is really just real-valued functions of real variables, and so we can ask whether these are continuously differentiable. The atlas defines a differential structure on , if every transition function is continuously differentiable.
More generally, for , the atlas is
said to define a differential structure, and is said to be
of class , if all the transition functions are -times
continuously differentiable, or real analytic in the case of
. Two differential structures of class on are
said to be isomorphic if the union of the corresponding atlases is
also a atlas, i.e. if all the new transition functions arising
from the merger of the two atlases remain of class . More
generally, two manifolds and are said to be
diffeomorphic, i.e. have equivalent
differential structure, if there
exists a homeomorphism such that the atlas of is
equivalent to the atlas obtained as -pullbacks of charts on .
The atlas allows us to define differentiable mappings to and from a manifold. Let
be a continuous function. For each we define
called the
representation of relative to chart , as the suitably
restricted composition
We judge to be differentiable if all
the representations are differentiable. A path
is judged to be differentiable, if for all differentiable functions , the suitably restricted composition is a differentiable function from to . Finally, given manifolds , we judge a continuous mapping between them to be differentiable if for all differentiable functions on , the suitably restricted composition is a differentiable function on .
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 |