Bundle Rank

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 .