Bundle Map

A bundle map is a map between bundles along with a compatible map between the base manifolds. Suppose  and  are two bundles, then

is a bundle map if there is a map  such that  for all . In particular, the fiber bundle of  over a point , gets mapped to the fiber of  over .

In the language of category theory, the above diagram commutes. To be more precise, the induced map between fibers has to be a map in the category of the fiber. For instance, in a bundle map between vector bundles the fiber over  is mapped to the fiber over  by a linear transformation.

For example, when  is a smooth map between smooth manifolds then  is the differential, which is a bundle map between the tangent bundles. Over any point in , the tangent vectors at  get mapped to tangent vectors at  by the Jacobian.