Transition function - Knowlpedia

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

\Phi_{ij}:=\Phi_i\circ \Phi_j^{-1}:U_{ij}\times F\longrightarrow U_{ij}\times F.

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

\Phi_{ij}(x,f)=(x,\,t_{ij}(x)(f)),

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

t_{ij}:U_{ij}\longrightarrow \mathrm{Diff}(F)

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.

Specified structure group

For a G-bundle, the transition maps are specified by maps $g_{ij}:U_i\cap U_j\to G$ acting on $F$ : $t_{ij}(x)(f)=g_{ij}(x)\cdot f$ . This extra structure records which changes of fiber coordinates are allowed.

Examples

Möbius line bundle: with two trivializations over overlapping arcs, the transition function is the constant map $t_{12}\equiv(-1)\in \mathrm{GL}(1,\mathbb{R})\subset \mathrm{Diff}(\mathbb{R})$ .
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 $p$ .
Vector bundles: in a vector bundle of rank $k$ , transition functions take values in $\mathrm{GL}(k,\mathbb{R})$ because trivializations are fiberwise linear.