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 R, and a complex line bundle has fibers isomorphic to C, but in both cases their rank is 1. The rank of the tangent bundle of a real manifold M is equal to the dimension of M. The rank of a trivial bundle M×R^k is equal to k. There is no upper bound to the rank of a vector bundle over a fixed manifold M.

bundle section | dimension | fiber | manifold | tangent bundle | vector bundle