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 {Uj} of B is an admissible atlas, and the transition functions gjk 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 gij = λi(x) λj–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: EB, 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) [S3-bundles over S4]; 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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