Transition function

Let $\pi:E\to M$ be a smooth fiber bundle with typical fiber $F$, and let $(U_i,\Phi_i)$ and $(U_j,\Phi_j)$ be local trivializations . On the overlap $U_{ij}=U_i\cap U_j$, the change of trivialization is the map

It is a diffeomorphism over $\mathrm{id}_{U_{ij}}$, so it necessarily has the form

where $t_{ij}(x)\in \mathrm{Diff}(F)$. The map

is the transition function (or transition map) of the two trivializations. It is a smooth map when $\mathrm{Diff}(F)$ is interpreted as “smoothly varying diffeomorphisms,” i.e. when the induced map $U_{ij}\times F\to F$, $(x,f)\mapsto t_{ij}(x)(f)$, is smooth.

Examples

1. Möbius line bundle: with two trivializations over overlapping arcs, the transition function is the constant map $t_{12}\equiv(-1)\in \mathrm{GL}(1,\mathbb{R})\subset \mathrm{Diff}(\mathbb{R})$.
2. Tangent bundle: if $(U_i,x)$ and $(U_j,y)$ are overlapping coordinate charts, then $t_{ij}(p)$ is given by the Jacobian matrix of the coordinate change $y(x)$ at $p$.
3. Vector bundles: in a vector bundle of rank $k$, transition functions take values in $\mathrm{GL}(k,\mathbb{R})$ because trivializations are fiberwise linear.