Transition Maps#
Definition: Transition Map
Let \((U_{\alpha}, \phi_{\alpha})\) and \((U_{\beta}, \phi_{\beta})\) be two charts on an \(n\)-manifold such that the intersection \(U_{\alpha} \cap U_{\beta}\) is non-empty.
The transition map from \((U_{\alpha}, \phi_{\alpha})\) to \((U_{\beta}, \phi_{\beta})\) is the function \(\tau_{\alpha \to \beta}: \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \phi_\beta(U_{\alpha} \cap U_{\beta})\) defined as the following composition:
Transition maps are transformations which allow us to change coordinates between two charts whenever they overlap somewhere. The way \(\tau_{\alpha \to \beta}\) does this is by first taking coordinates \(({p^1}_{\alpha}, \dotsc, {p^n}_{\alpha})\) w.r.t. \((U_{\alpha}, \phi_{\alpha})\) and mapping them to their corresponding \(p \in M\). Then it maps \(p\) to its coordinates \(({p^1}_{\beta}, \dotsc, {p^n}_{\beta})\) w.r.t. \((U_{\beta}, \phi_{\beta})\).
Theorem: Homeomorphicity of Transition Maps
Let \((U_{\alpha}, \phi_{\alpha})\) and \((U_{\beta}, \phi_{\beta})\) be two charts on an \(n\)-manifold such that the intersection \(U_{\alpha} \cap U_{\beta}\) is non-empty.
The transition map \(\tau_{\alpha \to \beta}: \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \phi_{\beta}(U_{\alpha} \cap U_{\beta})\) from \((U_{\alpha}, \phi_{\alpha})\) to \((U_{\beta}, \phi_{\beta})\) is a homeomorphism and its inverse is the transition map \(\tau_{\beta \to \alpha}: \phi_{\beta}(U_{\alpha} \cap U_{\beta}) \to \phi_{\alpha}(U_{\alpha} \cap U_{\beta})\) from \((U_{\beta}, \phi_{\beta})\) to \((U_{\alpha}, \phi_{\alpha})\).
Proof
TODO