The rank of a vector bundle is the dimension of its fiber. Equivalently, it is the maximum number of
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 
.
