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:
““There is Lowell, who’s striving Parnassus to climb
With a whole bale of isms tied together with rhyme,
He might get on alone, spite of brambles and boulders,
But he can’t with that bundle he has on his shoulders,
The top of the hill he will ne’er come nigh reaching
Till he learns the distinction ‘twixt singing and preaching;”
—James Russell Lowell (1819–1891)
“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)