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 
.
