Types of Fiber Bundles| 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: 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)
[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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