In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given.
... 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 ...
... The differential of a smooth map φ induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of M to the tangent bundle of N ... Equivalently (see bundle map), φ* = dφ is a bundle map from TM to the pullback bundle φ*TN over M, which may in turn be viewed as a section of the vector bundle Hom(TM,φ*TN) over M ...
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