Types
of Fiber Bundles |

**Differentiable Fibre Bundles**

$ __Def__: A fiber bundle
(*B*, *E*, *G*, *F*, *π*),
where *B*, *E*, *G*, *F* are differentiable manifolds, *π* is
a differentiable mapping, the covering {*U*_{j}}
of *B* is an admissible atlas, and the transition functions *g*_{jk} are
differentiable.

**Trivial Fiber Bundles**

* __Triviality Criteria__:

*- P*(*E*)
trivial iff *P*(*E*) admits a cross-section;

*- E* trivial
iff the transition functions can be written as *g*_{ij}
= *λ*_{i}(*x*) *λ*_{j}^{–1}(*x*);

*- P*(*E*) trivial implies *E* trivial;

*- B* contractible implies *E* trivial;

*- F* contractible implies *E* has a cross-section;

*- G* contractible implies *E* trivial.

* __Results__: All SU(2) bundles over 3-manifolds are trivial.

**Vector Bundles** > s.a. Jet Bundles; tangent bundles.

* __Idea__: A topological
space *E*, a continuous projection *π*: *E* → *B*,
and a vector space (over a field \(\mathbb K\)) structure on each fiber *π*^{–1}(*x*),
with local triviality, i.e., a fiber bundle with *F* = \(\mathbb K\)^{n} and *G* =
GL(*n*, \(\mathbb K\)).

@ __References__: in Milnor & Stasheff 74, ch 2–3.

> __Online resources__: see MathWorld page; Wikipedia page.

**Tensor Bundles**

> __Online resources__: see Encyclopedia of Mathematics page.

**Other Fiber Bundles and Additional Structure** > s.a. curvature; Hopf
Fibration; Jet; principal
fiber bundle; sheaf; Universal Bundle.

* __Triviality criteria__:
An *R*-bundle is trivial iff it admits *n* nowhere-dependent cross-sections.

@ __ General references__: Trautman RPMP(76)
[classification, and use in physics]; Crowley & Escher
DG&A(03)
[S^{3}-bundles
over S^{4}];
Lerman JGP(04) [contact fiber bundles].

@ __Generalizations__: Manton CMP(87)
[discrete bundles]; Brzeziński & Majid CMP(98) [coalgebra bundles]; Vacaru & Vicol
IJMMS(04)m.DG [higher-order,
and Finsler]; Bruce et al a1605-proc [graded bundles].

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