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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 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.
meaning that the fiber space S 1 (a circle) is embedded in the total space S 3 (the 3-sphere), and p : S 3 → S 2 (Hopf's map) projects S 3 onto the base space S 2 (the ordinary 2-sphere). The Hopf fibration, like any fiber bundle, has the important property that it is locally a product space.
The orthonormal frame bundle () of the Möbius strip is a non-trivial principal /-bundle over the circle. In mathematics , a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} .
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
Let : be a fiber bundle over a manifold with compact oriented fibers. If is a k-form on E, then for tangent vectors w i 's at b, let (, …,) = ()where is the induced top-form on the fiber (); i.e., an -form given by: with ~ lifts of to ,
The first definition of fiber space is given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. [ 3 ] [ 4 ] The theory of fibered spaces, of which vector bundles , principal bundles , topological fibrations and fibered manifolds are a special case, is attributed to Seifert , Hopf ...