Transition Maps

Definition: Transition Map

Let $(U_{\alpha },{\varphi }_{\alpha })$ and $(U_{\beta },{\varphi }_{\beta })$ be two charts for an $n$-manifold such that the intersection $U_{\alpha }\cap U_{\beta }$ is nonempty.

The transition map from $(U_{\alpha },{\varphi }_{\alpha })$ to $(U_{\beta },{\varphi }_{\beta })$ is the homeomorphism ${\tau }_{a\to b}:{\varphi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\varphi }_{\beta }(U_{\alpha }\cap U_{\beta })$ which maps the image of $U_{\alpha }\cap U_{\beta }$ under ${\varphi }_{\alpha }$ to its image under ${\varphi }_{\beta }$ by composing ${\varphi }_{\beta }$ and the inverse of ${\varphi }_{\alpha }$:

$${\tau }_{\alpha \to \beta }\stackrel{\text{def}}{=}{\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}$$

INTUITION

The transition map ${\tau }_{\alpha \to \beta }$ is simply a change-of-coordinates operation, which converts coordinates from the coordinate system ${\varphi }_{\alpha }$ to the coordinate system ${\varphi }_{\beta }$. However, since coordinates on a manifold are only locally defined, such a change of coordinates makes sense only for those points which have coordinates in both systems, i.e. those points which belong to $U_{\alpha }\cap U_{\beta }$.