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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.
The fibres of the bundle TP/G under the projection ρ carry an additive structure. The bundle TP/G is called the bundle of principal connections (Kobayashi 1957). A section Γ of dπ:TP/G→TM such that Γ : TM → TP/G is a linear morphism of vector bundles over M, can be identified with a principal connection in P.
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 ...
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.
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 ...
As an example, over the circle, there is a line bundle (i.e., a real vector bundle of rank 1) that is not isomorphic to a trivial bundle. This line bundle L is the Möbius strip (which is a fiber bundle whose fibers can be equipped with vector space structures in such a way that it becomes a vector bundle).