Local trivialization

A local trivialization of a fiber bundle $p:E\to B$ over an open set $U\subseteq B$ is a homeomorphism $p^{-1}(U)\cong U\times F$ commuting with the projections to $U$. For a vector bundle, it must be linear on each fiber. Compatible local trivializations specify the bundle structure and give its transition functions on overlaps.

Two local frames of a vector bundle differ on their overlap by an invertible matrix-valued function. These functions satisfy the Čech cocycle condition. For a holomorphic vector bundle they are holomorphic, and for a holomorphic line bundle they are scalar holomorphic units.

A complex vector bundle has structure group $G$ when it admits fiber-linear local trivializations whose transition matrices take values in $G$. Replacing frames by $h_i$ changes transitions to $h_i^{-1}{g}_{ij}{h}_j$ in the frame convention. Reducing the structure group means choosing new trivializations with values in a specified subgroup. The same transitions glue $U_i\times G$ by left multiplication to a right principal bundle.