Locally trivial fiber bundle

Example 2.3. — Let T be a Zariski torsor for a vector bundle over the smooth scheme X over S. Then the morphism T —> X is an A -weak equivalence. It follows from Lemma 2.16 applied to the class C of sheaves represented by smooth schemes over S which are affine (over Spec 2,)), Example 2.2 and the fact that any such torsor is trivial when the base is affine over Sped. More generaUy any smooth morphism Y X of schemes which is a locally trivial fibration in the Nisnevich topology with an A -contractible fiber is an A -weak equivalence. 

X SL fibre bundle with a fibre above is defined, which is locally trivial for all nonsero points of the disk The circle fibre (0,t),0 < t 1 is singular if a > 1. Now consider the manifold with points xi,... marked on it and encircle them with small disks. .., D. Then take the direct product P X and discard from it m full tori D X 5, 1 t m. In the place of these full tori glue in fibered full tori of the type (2.1). Denote the manifold obtained by P x5. It is a Seifert bundle and P is its base. 

We shall consider only locally trivial bimdles hence we will just say fiber bimdle or sometimes just bundle. In some texts one allows a slightly more general type of bundles, where the fibers may be different over various path-connected components of B. We shall not make use of this generality. 

The Born-Oppenheimer approximation, whose validity depends on there being a deep enough localized potential well in the electronic energy, has, however, been extensively treated. The mathematical approaches depend upon the theory of fiber bundles and the electronic Hamiltonian in these approaches is defined in terms of a fiber bundle. It is central to these approaches, however, that the fiber bundle should be trivial, that is that the base manifold and the basis for the fibers be describable as a direct product of Cartesian spaces. This is obviously possible with the decomposition choice made for O Eq. 2.42 but not obviously so in the choice made for O Eq. 2.43. 

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See also in sourсe #XX -- [ ]

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