Sunday, April 15, 2018

Bundle Section and Rank


A section of a fiber bundle gives an element of the fiber over every point in B. Usually it is described as a map s:B->E such that pi degreess is the identity on B. A real-valued function on a manifold M is a section of the trivial line bundle M×R. 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 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.

Source : WMW