Bundle rank is the dimension of a vector bundle's fiber. Equivalently, it is the maximum number
of pointwise linearly independent local bundle sections in a trivialization.
Naturally, the dimension here is measured in the appropriate
category. For instance, a real line
bundle has fibers isomorphic
with 
, and a complex line
bundle has fibers isomorphic
to 
, but in both cases their rank is 1.
The rank of the tangent bundle of a real manifold 
is equal to
the dimension of 
. The rank of a trivial bundle 
is equal
to 
. There is no upper
bound to the rank of a vector bundle over a
fixed manifold 
.
