Coordinate transition map

Definition

Let $(U,\varphi)$ and $(V,\psi)$ be coordinate charts on the same $n$ -dimensional manifold with $U\cap V\ne\varnothing$ . Their coordinate transition map, from $\varphi$ -coordinates to $\psi$ -coordinates, is

\psi\circ\varphi^{-1}: \varphi(U\cap V)\longrightarrow\psi(U\cap V).

It is a homeomorphism between open subsets of $\mathbb R^n$ . The charts are smoothly compatible exactly when this map is a diffeomorphism.

Role in an atlas

The transition maps record how local coordinate descriptions agree on overlaps. Pairwise smoothness of these maps is the compatibility axiom in a smooth atlas. Their derivatives give the transition functions of the tangent bundle.

Distinction from bundle transition functions

A coordinate transition map changes coordinates on the base manifold and is a map between open subsets of Euclidean space. A bundle transition function instead records how two local trivializations change fiber coordinates over the same base point.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: coordinate charts and transition maps.