Transition map
From Maths
Revision as of 06:20, 7 April 2015 by Alec (Talk | contribs) (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...")
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 also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
