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 set of
conditions for consistency is 
on 
, 
on 
, and 
on each triple overlap 
, where 
is the identity matrix.
A consistent set of transition functions for a vector bundle of bundle rank 
can be interpreted as an element
of the first non-Abelian Čech cohomology
pointed set 
. It generally has no natural group
structure because 
is non-Abelian for 
.
