Transition map

Definition

Given two charts $(U,\varphi)$ and $(V,\psi)$ on a topological $n-$manifold where $U\cap V\ne\emptyset$ a transition map allows us to move from local coordinates of $\varphi$ to local coordinates of $\psi$ as the picture on the right shows.

Transition map $\psi\circ\varphi$ on a topological $n$-manifold $M$

The transition map, $\tau$ is defined as follows:

$$\tau:\varphi(U\cap V)\rightarrow\psi(U\cap V)$$ given by $$\tau=\psi\circ\varphi^{-1}$$

$\tau$ is a Homeomorphism because both $\varphi$ and $\psi$ are homeomorphisms, making $\tau$ a chart, $(U\cap V,\tau)$

Extending to smooth structures

See Smoothly compatible charts

See also

• Chart
• Smoothly compatible charts
• Topological manifold