A transition function describes the difference in the way an object is described in two separate, overlapping coordinate charts,
where the description of the same set may change in different coordinates. This even
occurs in Euclidean space 
, where any rotation of the usual 
, 
, and 
axes gives another set of coordinates.
For example, on the sphere, person 
at the equator can use the usual directions of north, south,
east, and west, but person 
at the north pole must use something
else. However, both 
and 
can describe the region in between them in their coordinate charts. A transition
function would then describe how to go from the coordinate chart for 
to the coordinate chart for 
.
In the case of a manifold, a transition function is a map from one coordinate chart to another. Therefore, in a sense, a manifold is composed of coordinate charts, and the glue that holds them together is the transition functions. In the case of a bundle, the transition functions are the glue that holds together its trivializations. Specifically, in this case the transition function describes an invertible transformation of the fiber.
Naturally, the type of invertible transformation depends on the type of bundle. For instance, a vector bundle, which could be the tangent bundle, has invertible
linear transition functions. More precisely, a transition function for a vector
bundle of bundle rank 
, on overlapping coordinate charts 
and 
, is given by a function
 |
where 
is the general linear group. The fiber at
has two descriptions, and 
is the invertible
linear map that takes one to the other. The transition functions have to be consistent
in the sense that if one goes to another description of the same set, and then back
again, then nothing has changed. A necessary and sufficient condition for consistency
is the following: Given three overlapping charts, the product 
has to be the constant map to the identity
in 
.
A consistent set of transition functions for a vector bundle of bundle rank 
can be interpreted as an element of the first Čech
cohomology group of a manifold with coefficients in
.
