A section of a fiber bundle gives an element of the fiber over every point in . Usually it is described as a map such that is the identity on . A real-valued function on a manifold is a section of the trivial line bundle . Another common example is a vector field, which is a section of the tangent bundle.
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 .
Source : WMW