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This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles.
A real version of the Hopf fibration is obtained by regarding the circle S 1 as a subset of R 2 in the usual way and by identifying antipodal points. This gives a fiber bundle S 1 → RP 1 over the real projective line with fiber S 0 = {1, −1}. Just as CP 1 is diffeomorphic to a sphere, RP 1 is diffeomorphic to a circle.
A Lagrangian: given a fiber bundle ′, the Lagrangian is a function : ′. Suppose that the matter content is given by sections of E {\displaystyle E} with fibre V {\displaystyle V} from above. Then for example, more concretely we may consider E ′ {\displaystyle E'} to be a bundle where the fibre at p {\displaystyle p} is V ⊗ T p ∗ M ...
The Möbius strip can be constructed by a non-trivial gluing of two trivial bundles on open subsets U and V of the circle S 1.When glued trivially (with g UV =1) one obtains the trivial bundle, but with the non-trivial gluing of g UV =1 on one overlap and g UV =-1 on the second overlap, one obtains the non-trivial bundle E, the Möbius strip.
For a fiber bundle with structure group , the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems and are given as a -valued function on . One may then construct a fiber bundle F ′ {\displaystyle F'} as a new fiber bundle having the same transition functions, but possibly a different fiber.
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
At each point in the fiber , the vertical fiber is unique. It is the tangent space to the fiber. The horizontal fiber is non-unique. It merely has to be transverse to the vertical fiber. In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle.
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.