Difference between revisions of "Transition map"
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(Created page with " ==Definition== Given two charts {{M|(U,\varphi)}} and {{M|(V,\psi)}} on a topological {{M|n-}}manifold where {{M|U\cap V\ne\emptyset}}<ref>Introduction to smooth ma...") |
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==Extending to smooth structures== | ==Extending to smooth structures== | ||
| − | See [[Smoothly compatible]] | + | See [[Smoothly compatible charts]] |
==See also== | ==See also== | ||
* [[Chart]] | * [[Chart]] | ||
| − | * [[Smoothly compatible]] | + | * [[Smoothly compatible charts]] |
* [[Topological manifold]] | * [[Topological manifold]] | ||
Latest revision as of 06:33, 7 April 2015
Definition
Given two charts [ilmath](U,\varphi)[/ilmath] and [ilmath](V,\psi)[/ilmath] on a topological [ilmath]n-[/ilmath]manifold where [ilmath]U\cap V\ne\emptyset[/ilmath][1] a transition map allows us to move from local coordinates of [ilmath]\varphi[/ilmath] to local coordinates of [ilmath]\psi[/ilmath] as the picture on the right shows.
The transition map, [ilmath]\tau[/ilmath] is defined as follows:
[math]\tau:\varphi(U\cap V)\rightarrow\psi(U\cap V)[/math] given by [math]\tau=\psi\circ\varphi^{-1}[/math]
[ilmath]\tau[/ilmath] is a Homeomorphism because both [ilmath]\varphi[/ilmath] and [ilmath]\psi[/ilmath] are homeomorphisms, making [ilmath]\tau[/ilmath] a chart, [ilmath](U\cap V,\tau)[/ilmath]
Extending to smooth structures
See Smoothly compatible charts
See also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
