Transition map

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] 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.

Transition map [ilmath]\psi\circ\varphi[/ilmath] on a topological [ilmath]n[/ilmath]-manifold [ilmath]M[/ilmath]

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

• Chart
• Smoothly compatible charts
• Topological manifold