Bundle Map - Variants and Generalizations

Variants and Generalizations

There are two kinds of variation of the general notion of a bundle map.

First, one can consider fiber bundles in a different category of spaces. This leads, for example, to the notion of a smooth bundle map between smooth fiber bundles over a smooth manifold.

Second, one can consider fiber bundles with extra structure in their fibers, and restrict attention to bundle maps which preserve this structure. This leads, for example, to the notion of a (vector) bundle homomorphism between vector bundles, in which the fibers are vector spaces, and a bundle map φ is required to be a linear map on each fiber. In this case, such a bundle map φ (covering f) may also be viewed as a section of the vector bundle Hom(E,f*F) over M, whose fiber over x is the vector space Hom(E_x,F_f(x)) (also denoted L(E_x,F_f(x))) of linear maps from E_x to F_f(x).