Bundle Maps
It is useful to have notions of a mapping between two fiber bundles. Suppose that M and N are base spaces, and πE : E → M and πF : F → N are fiber bundles over M and N, respectively. A bundle map (or bundle morphism) consists of a pair of continuous functions
such that . That is, the following diagram commutes:
For fiber bundles with structure group G (such as a principal bundle), bundle morphisms are also required to be G-equivariant on the fibers.
In case the base spaces M and N coincide, then a bundle morphism over M from the fiber bundle πE : E → M to πF : F → M is a map φ : E → F such that . That is, the diagram commutes
A bundle isomorphism is a bundle map which is also a homeomorphism.
Read more about this topic: Fiber Bundle
Famous quotes containing the words bundle and/or maps:
“In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.”
—Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)
“And now good morrow to our waking souls,
Which watch not one another out of fear;
For love all love of other sights controls,
And makes one little room an everywhere.
Let sea-discoverers to new worlds have gone,
Let maps to other, worlds on worlds have shown,
Let us possess one world; each hath one, and is one.”
—John Donne (1572–1631)